UNIVERSITY    ^F       ,        KQRNIA 

DEPARTMENT   OF    CIVIL    ENGINEERING 

BERKELEY.  CALIFORNIA 


•engineering 


UNIVERSITY  OF  CALIFORNIA 
DEPARTMENT   OF   CIVIL    ENGINEERING 
,       BERKELEY.  CALIFORNIA 


MASONRY  DAM  DESIGN 


.. 
INCLUDING 


HIGH  MASONRY  DAMS 


BY 


CHARLES  E.  MORRISON,  C.E.,  PH.D. 

Formerly  of  the  Civil  Engineering  Department,  Columbia  University 


AND 


ORRIN   L.  BRODIE,  C.E. 

Mem.   Am.  Soc.   C.  E.,  Late  Asst.  Designing  Eng.,  N.  Y.   Board  of  Water  Supply 


SECOND  EDITION,  REVISED  AND  ENLARGED 

FIRST    THOUSAND 


NEW  YORK 

JOHN  WILEY  &  SONS,  INC. 

LONDON:    CHAPMAN  &  HALL,  LIMITED 

1916 


> 


Engineering 
Library 

Copyright,  1910,  1916 


BY 
CHARLES    E.   MORRISON  AND  ORRIN  L.  BRODIE 


THE  SCIENTIFIC  PRESS 

ROBERT  DRUMMOND  AND  COMPANY 

BROOKLYN.  N.  Y. 


PREFACE   TO   SECOND   EDITION 


IN  this  revision  the  authors  have  attempted  to  amplify 
some  of  the  features  which,  in  the  earlier  edition,  were 
little  more  than  touched  upon.  Chapters  on  the  Overfall 
and  Arched  types  of  masonry  dams  have  been  added, 
together  with  cross-sections  of  a  selected  series  of  masonry 
dams  chronologically  arranged.  The  last  are  for  the  pur- 
pose of  comparison  and  showing  the  development  of  the 
masonry  dam  from  the  time  of  the  massive  Spanish  type 
to  the  present. 

The  design  of  low  and  medium-sized  as  well  as  that  of 
high-masonry  dams,  may  be  prosecuted  according  to  the 
theory  and  methods  in  this  work,  as  general  expressions 
have  been  written  wherever  possible.  It  was  considered 
best  to  indicate  this  in  the  title  while  the  original  page 
captions  remain  the  same. 

The  authors  are  indebted  to  Miss  Bessie  N.  MacDonald, 
A.B.,  for  assistance  in  verifying  the  more  difficult  mathe- 
matical derivations  for  the  Arched  Dam.  Acknowledg- 
ment is  also  made  to  Mr.  Alfred  D.  Flinn,  Member  American 
Society  of  Civil  Engineers,  who  had  kindly  furnished  one 
of  the  authors  with  a  set  of  cross-sections  of  dams,  a  num- 
ber of  which  are  shown  in  the  series  in  Appendix  III. 

C.  E.  M. 

O.  L.  B. 

NEW  YORK  CITY, 
February,  1916. 

iii 


793985 


PREFACE   TO   FIRST  EDITION 


IT  is  the  practice  at  Columbia  University  to  require  of 
the  third-year  students  in  the  Department  of  Civil  Engineer- 
ing, the  execution  of  the  design  of  a  masonry  dam,  and 
to  aid  them  in  this  problem  they  have  heretofore  been 
furnished  with  "Notes  on  the  Theory  and  Design  of  High 
Masonry  Dams,"  prepared  some  years  ago  by  Prof.  Burr 
of  the  Department,  and  having  for  their  basis  the  method 
as  set  forth  by  Mr.  Edward  Wegmann. 

This  procedure  with  which  Wegmann  is  credited,  and 
which  was  developed  through  the  investigations  undertaken 
in  connection  with  the  Aqueduct  Commission  of  the  city 
of  New  York,  for  the  purpose  of  determining  a  correct 
cross-section  for  the  Quaker  Bridge  dam,  resulted  in 
the  first  direct  method  for  calculating  the  cross-section 
of  such  structures  and  is  essentially  a  development  of 
the  Rankine  theory. 

The  studies  appeared  first  in  the  report  made  by  Mr. 
A.  Fteley  to  the  chief  engineer  of  the  Aqueduct  Com- 
mission of  the  city  of  New  York,  dated  July  25,  1887, 
and  later  in  Mr.  Wegmann's  treatise  on  "The  Design 
and  Construction  of  Dams." 

Neither  in  the  report  nor  in  the  treatise  however, 
have  the  effects  of  uplift,  due  to  water  permeating  the 


VI  PREFACE 

mass  of  masonry,  and  of  ice  thrust,  acting  at  the  sur- 
face of  the  water  in  the  reservoir,  been  considered,  and 
in  consequence  of  this,  objection  might  be  legitimately 
raised  that  the  series  of  equations  determining  the  cross- 
section  fail  to  account  for  these  factors.  Some  difference 
of  opinion  may  exist  as  to  the  relative  importance  of 
these  considerations,  but  when  a  structure  of  great 
responsibility  is  projected,  conservatism  in  design  is 
essential. 

The  following  presentation  which  aims  to  supply  these 
omissions,  has  been  prepared  primarily  that  there  may  be 
had  in  convenient  form  a  text,  containing  the  general 
treatment  and  such  consideration  of  these  factors  as  more 
recent  practice  requires,  together  with  a  brief  statement 
regarding  the  late  investigations  undertaken  for  the  pur- 
pose of  determining  more  accurately  the  variation  of 
stress  in  masonry  dams. 

The  formulae  relating  to  uplift,  ice  thrust,  etc.,  were 
deduced  by  one  of  the  authors  and  have  been  used  in  part 
in  connection  with  the  design  of  the  large  dams  for  the 
new  water  supply  for  the  city  of  New  York. 

The  computations  for  the  design  of  a  high  masonry 
dam  are  appended  to  facilitate  the  ready  comprehension 
and  application  of  the  formulae. 

It  is  hoped  that  the  presentation  may  appeal  to  the 
practicing  engineer  as  well  as  the  student,  and  that  there 
may  be  found  therein  enough  to  compensate  him  for  the 

labor  involved  in  its  perusal. 

C.  E.  M. 

O.  L.  B. 

COLUMBIA  UNIVERSITY,  1910. 


TABLE  OF  CONTENTS 


CHAPTER  PAGE 

INTRODUCTION ix 

I.  UPWARD  PRESSURE  AND  ICE  THRUST i 

PART    I.  Upward  Pressure I 

PART  II.  Ice  Thrust 16 

II.  PRELIMINARY  CONSIDERATIONS 20 

Pressure  on  a  submerged  surface.     Center  of  pressure 20,  22 

Distribution  of  stress  in  a  masonry  joint 23 

III.  PART   I.  DEVELOPMENT  OF  FORMULA  FOR  DESIGN 31 

Nomenclature 31 

Conditions  for  stability 35 

Factors  of  safety 36 

Derivation  of  formulae 42 

PART  II.  DESIGN  FORMULA  A,  B,  C,  D,  E  AND  F  FOR  REFERENCE  53 

IV.  INVESTIGATION  FORMULAE — FOR  REFERENCE 69 

Formulae  for  position  of  the  resultant  on  horizontal  joint  or  base 
.    — Formulae  for  maximum  intensities  of  pressure  on  the  hori- 
zontal joints — Variation  in  assumed  value  for  hydrostatic  head 
causing  uplift  in  any  section. 

V.  THE  DESIGN  OF  "A  MASONRY  DAM 74 

VI.  WEIR  OR  OVERFALL  TYPE  OF  DAM 99 

VII.  THE  ARCH  DAM 134 

VIII.  RECENT  CONSIDERATIONS  OF  THE  CONDITION  OF  STRESS  IN  A 

MASONRY  DAM 169 

APPENDIX     I.  DERIVATION  OF  CANTILEVER  EQUATIONS 207 

APPENDIX  II.  MOVEMENTS  AND  STRESSES  IN  AN  ARCH  SUBJECTED  TO 
A  UNIFORM  RADIAL  LOAD  WITH  DERIVATION  OF  Equa- 
tion 8  FOR  ARCH  CROWN  DEFLECTION 221 

APPENDIX  III.  CROSS- SECTIONS  OF  EXISTING  MASONRY  DAMS 231 

INDEX 269 

vii 


« 
t 


INTRODUCTION 


THE  method  of  analysis  by  which  an  economical  cross- 
section  of  a  gravity  type  high  masonry  dam  may  be  most 
directly  calculated,  and  the  one  which  is  most  generally 
adopted  in  engineering  practice,  was  first  devised  by  Mr. 
Edward  Wegmann  through  studies  made  for  the  Aqueduct 
Commissioners  of  New  York  City,  in  connection  with  the 
design  of  the  New  Croton  Dam,  and  it  is  that  method 
which  will  be  employed  here,  though  it  will  receive  some 
modification  in  certain  particulars  and  be  elaborated  in 
certain  others. 

In  determining  the  cross-section  by  the  series  of  equa- 
tions developed  in  that  analysis,  no  account  is  taken  of 
uplift  due  to  water  penetrating  the  foundation  or  the  mass 
of  masonry  above,  nor  of  the  ice  thrust  acting  horizontally 
against  the  up-stream  face  of  the  dam,  at  the  surface  of 
the  water  in  the  reservoir,  though  reference  is  made  to  it. 
Present  practice  requires,  however,  that  these  two  factors 
be  recognized  where  a  structure  of  great  responsibility  is 
proposed,  and  in  this  respect  at  least  will  the  analysis 
be  amplified. 


IX 


MASONRY    DAM    DESIGN 

Including  High  Masonry  Dams 


CHAPTER   I 

UPWARD    PRESSURE  AND   ICE  THRUST 
PART  I — UPWARD  PRESSURE 

ALTHOUGH  it  had  been  appreciated  for  a  number  of 
years  that  a  complete  analysis  of  a  high  masonry  dam 
required  the  consideration  of  uplift  and  ice  thrust,  until 
comparatively  recently  no  structure  of  this  type  had  been 
designed  which  allowed  for  these  two  factors  in  the  com- 
putations. 

In  fact,  it  may  be  said  that  prior  to  the  year  1853 
masonry  dams  were  built  without  a  rational  consideration 
of  any  of  the  forces  acting  in  or  upon  them,  for  it  was  not 
until  then  that  de  Sazilly  first  indicated  the  principles 
upon  which  dam  design  is  based,  by  providing  for  a  suf- 
ficient safety  factor  against  sliding  and  overturning  and 
by  assigning  a  maximum  limit  of  pressure  against  the 
crushing  of  the  material. 

Some  time  later  Rankine  added  to  the  theory  by  pre- 
scribing the  well-known  requirement  that  the  line  of 
resultant  pressure  for  reservoir,  full  or  empty,  should  lie 
within  the  middle  third  of  the  structure,  to  preclude  the 
possibility  of  tension  in  any  joint,  and  suggested  that  the 


2  HIGH   MASONRY    DAM    DESIGN 

'e^t-KlJ^ 

limit  of  pressure  should  be  made  less  for  the  down-stream 

edge  than  for  the  up-stream. 

In  1884,  when  the  Aqueduct  Commission  of  New  York 
City  came  to  design  the  New  Croton  Dam,  a  structure 
between  275  and  300  feet  high,  then  the  highest  in  the 
world,  and  exceeding  the  next  highest  by  about  100  feet, 
it  was  found  necessary  to  modify  some  of  the  older  con- 
clusions with  respect  to  dam  design,  in  order  to  make 
the  theory  applicable  to  their  particular  problem. 

Thus,  where  heretofore  the  prescribed  limit  of  crushing 
strength  of  masonry  had  been  assumed  to  be  between 
6  and  10  tons  per  square  foot,  they  increased  it  to  16 
tons,  as  it  had  been  demonstrated  that  such  pressures 
actually  existed  in  dams  still  doing  duty,  and  since,  with 
the  lower  values,  the  computations  would  have  given  a 
horizontal  face  at  a  joint  300  feet  below  the  top.  Both 
upward  pressure  and  ice  thrust  were  considered,  but  both 
in  turn  were  disregarded.  The  former  because  it  was 
felt  that  the  condition  of  the  masonry  and  of  the  founda- 
tion was  such  that  the  entering  of  water  would  be  a  remote 
possibility,  and  the  latter  because  it  was  believed  that 
the  mass  of  the  masonry  was  sufficiently  great  to  care  for 
any  additional  forces  due  to  the  ice  thrust. 

It  remained,  therefore,  for  the  engineers  of  the  Wa- 
chusett  dam  in  Massachusetts  to  be  the  first  in  the  United 
States  actually  to  incorporate  uplift  and  ice  thrust  in  the 
design  of  a  high  masonry  dam.  They  may  have  been  led 
to  this  precaution  by  the  fact  that  the  structure  was 
located  only  one -half  mile  above  a  town  of  some  13,000 
inhabitants,  where  a  failure  would  result  in  enormous 
loss  of  life,  and  where  it  was,  in  consequence,  necessary 


HIGH    MASONRY    DAM    DESIGN  3 

to  be  particularly  careful,  but  at  any  rate  they  were  the 
first  to  adopt  these  l;wo  considerations  in  the  design  of  a 
high  masonry  dam. 

Their  assumption  for  uplift  was  two-thirds  of  the 
static  head  at  the  up-stream  edge,  diminishing  as  a  straight 
line  to  zero  at  the  down-stream  edge,  and  for  ice  thrust, 
47,000  pounds  per  linear  foot  of  dam,  or  equivalent  to 
the  crushing  strength  of  ice  one  foot  thick. 

To-day,  in  the  light  of  experience,  no  structure  of 
this  character  would  be  built  without  careful  consideration 
of  both  these  elements,  and  it  is  doubtful  if,  under  any 
circumstances,  they  would  be  eliminated  entirely,  though 
they  might  not  receive  the  same  weight  they  were  given 
in  the  computations  for  the  Wachusett  Dam. 

That  engineers  are  not  fully  agreed  on  the  matter 
of  uplift  and  ice  thrust  and  that  a  considerable  diversity 
of  opinion  exists  in  the  profession  with  respect  to  them, 
may  perhaps  be  partially  explained  by  the  fact  that  the 
former  does  not  lend  itself  to  an  exact  treatment,  while, 
with  regard  to  the  latter,  there  are  no  exact  data  as  to 
the  expansive  force  of  ice  acting  at  the  surface  of  a  res- 
ervoir. Furthermore,  there  are  many  high  masonry  dams 
now  standing  which  were  designed  with  no  consideration 
being  given  to  these  two  factors,  and  this  would  seem 
to  refute  the  argument  that  they  are  necessary  consid- 
erations for  safety. 

It  is  recognized,  however,  that  the  influence  of  upward 
pressure  and  ice  thrust  on  the  stability  of  masonry 
dams,  together  with  the  actual  internal  distribution 
of  stress  in  very  large  masses  of  masonry  are  probably 
the  most  indefinite  factors  in  the  design  of  such  structures. 


4  HIGH    MASONRY    DAM    DESIGN 

Uplift. — It  may  not  be  out  of  place  to  explain  here 
at  some  length  what  "  uplift  "  means  and  how  it  may  be- 
come active. 

In  a  masonry  mass,  especially  a  large  concrete  mass, 
cracks  can  be  formed  by  temperature  changes,  due  to  the 
setting  of  the  concrete  in  the  first  place,  and  to  sub- 
sequent daily  and  seasonal  exterior  temperature  varia- 
tions. 

Contraction  joints,  provided  to  meet  the  effect  of  such 
temperature  changes  in  the  body  of  the  masonry,  are  often 
built  in  large  dams. 

Discontinuity  of  the  mass  of  a  large  masonry  structure 
like  a  dam,  owing  to  interruption  and  resumption  of  con- 
struction work  from  day  to  day,  is  also  evidenced  by 
joints,  mostly  horizontal,  perhaps,  but,  in  spite  of  the 
utmost  attempts  to  preserve  continuity,  often  unavoid- 
able. 

Temperature  cracks,  contraction  and  construction  joints, 
then,  all  tend  to  affect  permeability  to  a  greater  or  less 
degree,  admitting  water  to  the  body  of  a  dam  according 
to  the  pressure  exerted  by  that  water. 

Besides,  as  it  has  been  observed  that  water  under 
sufficient  head  has  passed  through  30  feet  thickness  of 
good  concrete  and  that  under  enormous  pressures  water 
has  been  made  to  ooze  through  cast-steel  cylinders,  it  may 
be  appreciated  from  the  above  considerations  that  water 
from  a  reservoir  may  enter  the  masonry  mass  of  the 
dam.  In  fact,  it  has  been  frequently  found  in  high  masonry 
dams  that,  following  construction  and  upon  filling  the  res- 
ervoir, small  issuing  streams  or  leaks  have  appeared  on  the 
down -stream  face.  These  leaks  have  been  observed  at 


HIGH    MASONRY    DAM    DESIGN  5 

> 

the  base,  down-stream,  as  wfcll  as  higher  up,  so  it  may  be 
safely  assumed  that  water  from  the  reservoir  may  pene- 
trate the  natural  foundations  as  well  as  the  masonry 
above,  especially  if  the  former  consist  of  a  porous  or 
stratified  formation. 

Therefore,  in  a  structure  of  considerable  height,  which 
retains  a  body  of  water  behind  it,  there  may  be  exerted 
a  powerful  vertical  force  acting  upward  under  the  dam 
or  on  some  joint  of  the  masonry  above.  This  force  is 
commonly  termed  "  uplift,"  and  will,  of  course,  depend, 
for  its  amount,  upon  the  hydrostatic  head.  Furthermore, 
this  force,  due  to  the  water  pressure,  tends  to  counter- 
balance the  downward,  vertical  component  of  all  forces 
acting  in  or  upon  the  structure. 

Obviously,  were  this  "  uplift  "  to  become  sufficiently 
great  it  might  actually  float  the  structure  off  its  foundation 
or  off  any  joint,  whereupon  the  horizontal  water  thrust 
back  of  the  dam  would  complete  the  destruction  by 
sliding  it  down-stream. 

Upward  pressure,  therefore,  should  receive  considera- 
tion, both  from  the  standpoint  of  its  effect  in  the  foundation 
of  the  dam,  and  also  in  any  of  the  joints  above. 

Naturally,  it  is  much  more  difficult  to  ascertain  the 
condition  of  this  with  respect  to  the  foundation,  as  the 
latter 's  physical  characteristics  are  never  revealed  until 
actual  work  has  begun  on  the  structure  and  the  site  is 
uncovered.  For  this  reason,  it  should  be  made  imperative 
to  examine  by  exploration,  drill  holes,  etc.,  as  completely 
as  possible,  the  nature  of  the  foundation,  so  that  its  true 
state  may  be  at  least  approximately  known,  and  so  that, 
also,  proper  provision  for  upward  pressure  may  be  made. 


6  HIGH    MASONRY    DAM    DESIGN 

An  examination  of  the  outcropping  rock  at  a  dam  site 
will  never  be  sufficient  to  determine  the  nature  of  the 
foundation  below,  as  the  latter  may  not  conform  with  the 
exposed  surface.  Core  borings  -should  be  made  at  frequent 
intervals.  They  should  be  driven  well  into  the  bed  rock 
to  develop  the  character,  and  where  there  is  limestone  with 
a  likelihood  of  cavities,  particular  care  should  be  exercised. 
Upon  such  cavities  being  uncovered,  they  should  be  filled 
with  grout  and  concrete,  so  as  to  preclude  the  entrance 
of  water.  Very  porous  sandstone  or  the  existence  of 
seams  and  strata  may  give  rise  to  very  dangerous  con- 
ditions. Thus,  an  examination  by  borings  at  the 
Austin,  Pa.,  dam  site  would  have  indicated  the  porosity 
of  the  rock,  and  might  have  been  the  means  of  prevent- 
ing the  disaster  which  followed.  All  foundations  should 
be  tested  for  tightness  by  applying  air  or  water  pressure 
to  the  drill  holes. 

Even  in  the  best  foundation,  however,  it  may  be  said 
that  there  is  no  absolutely  water-tight  condition. 

All  water  getting  into  the  dam  should  be  collected  in 
a  chamber  or  tunnel,  carried  outside  and  measured  for 
quantity.  Thereby  a  measure  of  the  water-tightness  of 
the  dam  may  be  ascertained. 

Treatment  of  Uplift.  —  There  is  not  the  clearest  con- 
ception among  engineers  as  to  how  to  allow  for  this  up- 
ward pressure,  but  in  some  of  the  more  recent  discussions,  * 
it  has  been  suggested  that  perhaps  three  general  con- 
ditions may  be  recognized. 


*  "  Provision  for  Uplift  and  Ice  Pressure  in  Designing  Masonry  Dams." 
By  C.  L.  Harrison,  Trans.  Am.  Soc.  C.E.,  Vol.  LXXV,  p.  142. 


HIGH   MASONRY   DAM   DESIGN  7 

1.  The    case   where   no  ^upward   pressure    could   exist 
because   the   foundation  rock,  and  the  joints  of  the  dam, 
were  so  tight  that  no  water   could  possibly  enter.      Evi- 
dently for  such  a  condition  provision  for  upward  pressure 
in  the  design  of  the  dam  is  unnecessary. 

2.  Second,   the  case  at  the  other  extreme,  where  the 
rock  is  of  such  a  nature  that  water  may  freely  enter  the 
foundation,  and  as  freely  leave  it  from  the  lower  edge  of 
the  dam.     Here  it  is  quite  evident  that  the  water  would 
enter  with   the  full  hydrostatic  pressure   acting   at   that 
point  or  elevation,   and  if  the  water  flowed  away  freely 
from  the  down-stream  edge,  the  hydrostatic  head  would 
be  zero  at  this  latter  point.     It  might  be  a  fair  assump- 
tion to  conclude  that  the  pressure  varied  as  a  straight  line 
between   the   up-stream   and   down-stream   edge,*    which 
would   give  an  equivalent  pressure  over  the  entire  base 
of  one-half  the  hydrostatic  head  assumed  acting  at  that 
point. 

3.  The  third  case  might  be  represented  by  that  which 
would    be  intermediate  between  cases  i  and  2 ;    in  other 
words,  where   there  was   easy   access   to   the   foundation, 
but  not  such  easy  access  from  it.     Under  these  conditions 
the  pressure  at  the  heel  would  be  assumed  equal  to  the 
hydrostatic  head,  while  at  the  toe  it  would  be  equal   to 
that  pressure  represented  by  the  head  of  the  issuing  stream. 

It  therefore  becomes  a  question  for  the  engineer  to 
decide,  from  a  knowledge  of  the  condition  of  the  founda- 
tion, as  to  what  degree  of  entering  water  and  consequent 

*  Cf.  Proceedings  Am.  Soc.  C.E.  for  May,  1915,  "  Experiments  on  Uplift." 
These,  however,  are  upon  too  small  a  scale  to  yield  conclusions  other  than 
those  applying  to  the  experiments  themselves. 


8  HIGH    MASONRY    DAM    DESIGN 

uplift  may  exist,  and  to  provide  for  it  accordingly.  It 
is  just  because  the  matter  is  based  upon  judgment  that 
such  a  diversity  of  opinion  prevails. 

Generally  it  is  quite  likely  that  none  of  the  above 
conditions  will  strictly  apply,  but  rather  varied  combina- 
tions of  them,  so  that  it  becomes  difficult  to  conclude  how 
to  dispose  of  this  question. 

Some  engineers  demand  that  the  structure  shall  be 
designed  for  the  full  static  head  acting  over  the  entire 
base,  while  others  advise  that  no  allowance  whatsoever 
be  made,  but  it  is  generally  conceded  that  some  dams, 
dependent  on  the  kind  of  their  foundations,  need  provision 
for  uplift. 

As  an  example  of  the  former,  there  may  be  cited  the 
dam  at  Marklissa,  Prussia,  over  the  Queis,  *  while  the  New 
Croton  Dam  is  an  example  of  a  very  important  structure 
of  this  type  where  such  provision  was  absolutely  elim- 
inated in  the  design. 

There  are  several  ways  in  which  upward  pressure  may 
be  cared  for:  First,  by  adding  a  sufficient  section  to  the 
dam  to  offset  the  upward  pressure,  and  second,  by  providing 
drainage  wells  and  galleries  to  intercept  all  entering  water, 
carrying  it  away  through  a  discharge  gallery,  or  conduit, 
to  the  lower  side  of  the  dam,  and  at  the  same  time  by 
carefully  providing  for  as  impervious  an  up-stream  face 
as  possible. 

In  the  foundation  an  adequate  cut-off,  of  width  and 
depth  determined  by  examination  of  conditions  disclosed 
during  the  progress  of  foundation  excavation,  is  often 

*  Trans.  Am.  Soc.  C.E.,  Vol.  LXXV. 


HIGH    MASONRY    DAM    DESIGN  9 

» 

advisable.  The  exploratory  borings  should  usually  indi- 
cate beforehand  this  necessity,  so  that  the  final  extent  to 
which  a  cut-off  trench  is  taken  remains  to  be  decided  during 
its  actual  excavation.  Borings  may  be  extended  from  its 
faces  and  bottom  to  reach  seams  and  pockets  to  be  filled 
by  grouting  under  pressure  before  the  concrete  of  the 
cut-off  trench  is  placed. 

The  drainage  wells,  slightly  inclined  to  the  vertical, 
and  the  cut-off  are  placed  as  near  as  consistent  to  the 
up-stream  face  of  the  dam,  and  galleries  are  built  longi- 
tudinally to  the  up-stream  face.  While  the  wells  and  gal- 
leries may  be  nearly  completely  effective  in  intercepting 
percolation,  they  cannot  be  considered  absolutely  so,  and 
consequently  may  allow  some  water  to  get  down-stream. 
Such  seepage  would  then  result  in  upward  pressure  down- 
stream from  the  wells  and  galleries,  and  if  the  water  had 
connection  in  any  way  with  the  reservoir,  pressure  on  a 
joint  due  to  the  full  static  head  might  result.  These  last 
remarks  apply,  but  with  less  force,  perhaps,  to  the 
foundation  cut-off. 

The  theory  of  this  intercepting  drainage  system  is 
that  any  water  having  gotten  into  the  dam  due  to  the 
static  pressure  acting  on  faults  or  cracks  in  the  com- 
paratively more  impervious  up-stream  face,  will  be 
caught  and  prevented  from  going  any  further  into  the 
structure. 

It  should  not  be  assumed  that  because  of  an  impervious 
up-stream  face,  and  because  of  drains  and  cut-offs,  no 
water  reaches  the  body  of  the  dam  below  the  latter,  for 
there  may  be  construction  joints,  and  contraction  cracks 
in  the  face,  and  in  places  the  cut-offs  and  the  down- 


10  HIGH    MASONRY    DAM    DESIGN 

stream  portion  may  be  less  pervious  than  the  up-stream 
part,  as  a  consequence  of  which  uplift  may  exist  in  the 
latter. 

Additional  means  of  protection  should  be  provided  in 
the  form  of  drainage  channels  to  lead  the  water  away  from 
the  down -stream  portion  of  both  the  foundation  and  the 
body  of  the  dam. 

As  an  example  of  intercepting  drains,  at  the  Cataract 
Dam,  which  furnishes  the  water  supply  for  Sydney,  Australia, 
"  the  upper  face  of  masonry  was  built  with  special  care 
to  a  depth  of  2  or  3  feet,  and  this  alone  is  relied  upon 
to  prevent  seepage.  The  rest  of  the  dam  is  built  of  good, 
though  more  pervious,  masonry,  and  throughout  the  whole 
were  placed  6-inch  rectangular  conduits  filled  with  broken 
stone  parallel  to  and  about  6  feet  back  from  the  up-stream 
face.  These  are  collected  into  6-irich  earthenware  pipes, 
laid  at  right  angles  to  the  longitudinal  axis  of  the  dam, 
with  exits  on  the  down-stream  face."  *  (Cf.  cross-section, 
Olive  Bridge  Dam,  page  96.) 

These  systems  of  drainage  naturally  tend  to  eliminate 
upward  pressure  and  consequently  increase  the  stability 
of  the  dam,  and  would  seem  justifiable  in  the  case  of  all 
important  structures. 

In  small  dams  drainage  wells  are  not  so  easily 
provided  and  the  protection  is  relatively  less  complete, 
because  there  is  a  certain  minimum  distance  from  the 
upper  face  within  which  the  drains  cannot  well  be  extended. 

With  the  correction  for  upward  pressure  applied  in 
the  form  of  increased  section,  the  water  entering  the  dam 
is  wasted,  which  is  a  considerable  item  of  cost,  while  in 
*  Mr.  Allen  Hazen,  Trans.  Am.  Soc.  C.E.,  Vol.  LXXV,  p.  154. 


HIGH    MASONRY    DAM    DESIGN 


11 


addition  the  cost  of  the  Structure  is  increased  by  the 
added  masonry.  With  collecting  galleries  both  of  these 
items  may  be  partially  eliminated.  The  cut-off  wall  in 
the  foundation  will  often  prove  an  economy  in  this  respect. 
The  question  of  cost  may  become  a  very  important 
one,  as  in  the  case  of  hydro-electric  developments  where 
additional  cross -section  may  mean  such  an  increase  in  the 
cost  as  to  cause  the  abandonment  of  the  project. 

TABLE  I 
WIDTH  OF  BASE  OF  DAM  GIVEN  IN  FEET  FOR  VARYING  HEIGHTS 


Height  of  dam  in  feet  

5 

IO 

30 

60 

IOO 

250 

Water    pressure    only   (hori- 

zontal)   

1>.2 

6.4 

IQ 

4° 

65 

161 

Uplift  as  described  below.  .  .  . 

4.2 

8.4 

25 

5i 

84 

211 

The  effect  of  uplift  in  its  tendency  to  increase  the  mass 
of  masonry  is  shown  in  Table  I,  for  six  triangular  dam 
cross-sections,  where  the  upward  pressure  is  assumed 
equal  to  the  static  head  at  the  heel,  and  diminishing  as 
a  straight  line  to  zero  at  the  toe.  For  convenience,  also 
the  width  at  the  top  is  taken  equal  to  zero.* 

Table  II  and  Fig.  i  represent  five  cross-sections  of 
dams,  four  of  which  were  directly  designed  for  comparative 
purposes,  showing  the  effect,  not  only  of  uplift,  but  of 
ice  pressure,  upon  the  top  and  bottom  dimensions,  super- 
elevation necessary,  as  well  as  the  comparative  volumes 
resulting.  These  differ  from  those  previously  cited,  in  that 
the  full  hydrostatic  heads  for  dams  sustaining  different 
heads  are  there  employed  for  "  uplift  "  comparison, 


*  Mr.  W.  J.  Douglas  in  Trans.  Am.  Soc.  C.E.,  Vol.  LXXV,  p.  207. 


12 


HIGH    MASONRY   DAM    DESIGN 


TABLE   II 
DIMENSIONS,  CONDITIONS,  ETC.,  FOR  FIVE  CROSS-SECTIONS  OF  DAMS. 


Super- 

Percent- 

Conditions of 

elevation 

A  __„ 

age  of 

Cross- 
sections. 

loading. 
(Masonry,  140 
pounds  per 
cu.ft.) 

Top 
width, 
in  feet, 

above 
normal 
reservoir 
surface, 

Base, 
in 
feet. 

x\reat 
in 
square 
feet. 

excess  of 
area  over 
the  area 
of  D. 

in  feet. 

Austin,  Pa. 

2    5 

2    5 

"}O 

840  ± 

o± 

A 

Ice,  *    uplift,    and 

o 

O 

O 

V^W  _J_ 

horizontal  water 

thrust.  ........ 

10.  0 

IO.O 

4i-5 

1475 

76 

£t  

Uplift  and  horizon- 

tal water  thrust. 

6.0 

5-o 

40.0 

992 

19 

ct  

Uplift  and  horizon- 

tal water  thrust. 

IO.O 

IO.O 

39-5 

999 

2O 

D  

Horizontal  water 

thrust  

5-o 

5.0 

33-1 

836 

0 

*  21,500  pounds  per  lin.  ft.  of  dam. 

t  B  and  C  are  subjected  to  the  same  conditions  of  loading.      Flood  level  at  +2.5  ft. 
The  resultant  on  any  base  or  horizontal  joint  of  masonry  is  at  the  down-stream  "  middle 
third"  point. 

while  but  one  depth  of  reservoir  is  employed  in  the  latter 
set  of  comparisons. 

The  uplift  intensity  is  assumed  as  varying  uniformly 
from  a  maximum  at  the  heel  to  zero  at  the  toe  and  the  uplift 
is  considered  as  acting  over  only  a  portion  of  the  area  of 
the  joint.  This  is  cared  for  by  assuming  only  a  portion 
of  the  full  hydrostatic  head  as  acting  at  the  up-stream  end 
of  the  joint  considered.  Two-thirds  of  the  full  up-stream 
head  is  used. 

Extent  and  Distribution. — Upward  pressure  cannot  act 
over  the  entire  area  of  a  joint  or  base,  or  the  dam  would 
be  floating.  Total  pressure  on  the  parts  of  the  joint  in 
contact  must  equal  the  difference  between  the  weight  of 
dam  and  uplift,  and  be  less  than  the  crushing  strength 


HIGH    MASONRY    DAM    DESIGN 


13 


of  the  material  or  not  morA  than  500  pounds  per  square 
inch.     "  The  assumption  that  only  one-third  of  the  rock 


15.0 


-47.5 


faces  are  in  direct  contact  gives  pressure  as  great  as  prob- 
ably occur,  and  leads  to  the  conservative  assumption  of 


14  HIGH   MASONRY   DAM    DESIGN 

upward  pressure  over  two-thirds  of  the  base,  and  varying 
according  to  the  resistance  losses  in  passing  through  the 
rock  or  masonry."  Hence  a  uniformly  varying  intensity 
is  usually  employed.  "  Where  upward  pressure  must  be 
allowed  for  in  the  base,  there  is  no  economy  in  failing  to 
allow  for  it  in  the  joints  above  the  base." 

A  distinction  should  be  drawn  between  the  uplift 
conditions  which  may  be  encountered  in  the  foundations 
and  those  higher  up  in  the  dam,  and  the  foregoing  as- 
sumptions in  regard  to  uplift  may  be  modified  for  special 
cases.  For  example,  a  trapezoidal  (instead  of  a  tri- 
angular) distribution  of  intensity  due  to  uplift  may  be 
found  advisable  for  a  foundation.  In  cases  where  inter- 
cepting drainage  wells  are  provided  in  the  body  of  the 
masonry,  as  for  the  Olive  Bridge  and  Kensico  Dams,  it 
would  be  reasonable  to  assume  a  triangular  disposition 
of  intensities,  with  the  maximum  at  the  heel  as  before, 
but  running  out  to  zero  at  or  a  little  beyond  the  line  of 
wells. 

It  has  been  suggested  as  reasonable  to  assume  that 
if  water  is  to  be  properly  excluded  from  a  masonry  dam, 
the  same  general  methods  should  be  applied  as  are  em- 
ployed in  waterproofing  any  foundation,  such  as  the  use 
of  several  layers  of  tarred  felt,  a  waterproofing  surface 
coat  of  some  kind,  or  by  pouring  wet  concrete  continuously. 
Objection  to  the  first  method  would  be  that  such  foreign 
substance,  in  layers,  would  form  a  plane  of  cleavage  that 
would  defeat  its  very  purpose  by  providing  a  weak,  hori- 
zontal joint. 

It  is  known  that  "  joints  between  two  successive  days' 
work  in  concrete  may  become  planes  of  entry  for  the  water." 


HIGH    MASONRY   DAM    DESIGN 


15 


Furthermore,  when  mortar  4s  used  in  ashlar  or  rubble 
masonry  it  must  be  dry  enough  to  handle,  and  conse- 
quently it  lacks  the  wet  consistency  necessary  to  make 
it  waterproof,  thus  permitting  water  to  enter  at  the 
joints. 

As  a  general  proposition,  then,  upward  pressure  and 
ice  thrust  may  be  said  to  be  more  or  less  dependent  upon 
local  conditions,  as  to  the  extent  to  which  they  should  be 
considered  in  any  given  case. 

Three  cases  of  failure,  undoubtedly  due  in  part  to 
upward  pressure  for  which  no  allowance  was  made  in 
the  design,  may  be  instanced,  but  at  the  same  time  it  should 
be  pointed  out  that  all  of  these  dams  were  established  on 
poor  foundations. 


Maximum 
Height. 

Built. 

Failed. 

Bouzey  Darn 

,  France  

72 

1878-81 

April  27    180^ 

Austin  Darn 

Texas. 

68 

I  80  I—  Q2 

April    7   1  900 

Austin  Dam, 

Pa  

so 

IQOQ-IO 

Seot.  1*0  ion 

In  the  Bouzey  Dam  the  -foundation  was  on  fissured 
red  sandstone,  and  quite  permeable,  and  the  excavation 
was  carried  down  to  only  a  fairly  good  bottom,  and  by 
no  means  to  solid  rock.  The  foundation  of  the  Austin, 
Texas,  dam  was  located  partly  over  a  fault  75  feet  wide, 
filled  with  adobe  with  occasional  streaks  of  red  clay,  nor 
was  the  foundation  trench  excavated  deep  enough,  while 
the  protection  on  the  downstream  side  was  insufficient. 
The  Austin,  Pa.,  dam  was  founded  on  sandstone,  underlaid 
by  shale  having  fissures  filled  with  clay,  sand,  and  gravel. 


16  HIGH   MASONRY   DAM   DESIGN 

PART  II — ICE  THRUST 

It  is  realized  that,  in  countries  where  low  tempera- 
tures prevail  in  winter,  the  pressure  of  expanding  ice 
against  the  face  of  a  dam  at  the  level  of  the  water  sur- 
face may  become,  before  crushing  of  the  ice  takes  place, 
a  very  tangible  force  tending  to  destroy  the  equilibrium 
of  forces  and  thus  overturn  the  structure.  This  expan- 
sion will  occur  when  the  ice  is  formed  under  low  tempera- 
tures and  when  higher  temperatures  later  prevail. 

In  addition  to  this,  the  ice  may  deliver  a  considerable 
blow  or  impact  when  it  has  formed  into  floes  and  the 
wind  carries  it  down  to  the  face  of  the  dam.  In  this  latter 
connection,  however,  it  is  well  to  remember  that  under 
the  force  of  the  wind  the  jagged  points  of  the  ice  floes 
would  first  come  into  contact  with  the  dam,  and  these 
would  be  broken  off.  But  under  any  conditions  the 
expansive  force  of  the  ice  will,  without  question,  be  the 
more  important  consideration. 

It  is  to  guard  against  these  forces  which  produce  an 
additional  tendency  to  destroy  the  equilibrium  of  the 
structure,  that  ice  thrust  is  considered. 

Where  the  reservoir  has  sloping  sides  it  would  seem 
reasonable  to  assume  that  the  expanding  ice  would  tend 
to  slide  up  the  shores,  and,  as  the  face  of  the  dam 
is  only  a  small  part  of  the  shore  line,  that  there  would 
be  comparatively  little  force  exerted  against  it.  If  the 
walls  of  the  reservoir  were  vertical,  however,  this  would 
not  apply. 

In  very  cold  climates  it  may  not  be  wise  nor  safe  to 
assume  that  the  ice  may  be  kept  clear  of  the  dam  by  means 


HIGH    MASONRY    DAM    DESIGN  17 

» 

of  maintaining  an  o^eri  trench  next  to  the  upstream  face, 
for  on  the  contrary  the  full  pressure  may  be  exerted. 

In  the  Cross  River  Dam  and  in  the  Croton  Falls  Dam, 
recently  erected  on  the  watershed  of  the  Croton  River  near 
the  new  Croton  Reservoir,  and  with  no  inhabitants  imme- 
diately below  them,  besides  allowing  for  upward  pressure, 
ice  thrust  was  figured  at  24,000  pounds  per  linear  foot  for 
the  former  and  30,000  pounds  per  linear  foot  for  the  latter. 

However,  in  this  connection,  it  should  be  borne  in  mind 
that  reservoirs  for  domestic  supply  are  usually  drawn 
down  during  the  period  when  ice  prevails,  and  that  a  point 
of  application  of  the  ice  thrust  thus  results  much  lower 
than  might  be  assumed,  which  the  structure  is  better 
able  to  resist.  But  if  a  storage  reservoir  is  to  be  kept  at 
high  level  during  the  ice  season,  full  pressure  should  be 
considered  acting  at  the  top. 

The  late  C.  L.  Harrison  *  concluded  that  under  the 
following  conditions  it  was  not  necessary  to  provide  for 
ice  pressure : 

"(i)  For  the  ordinary  storage  reservoir  with  sloping 
banks,  in  climates  where  the  maximum  thickness  of  ice 
is  6  inches  or  less — for  dams  with  southern  exposure  this 
limit  may  be  placed  as  high  as  i  foot. 

"  (2)  For  reservoirs  which  are  filled  during  the  flood 
season  and  from  which  all  the  stored  water  is  drawn  off 
each  year  during  the  low- water  season.'  This  would  in- 
clude even  the  large  reservoirs  on  the  head-waters  of  the 
Mississippi  River,  where  the  ice  has  a  thickness  of  more 
than  4  feet,  and  the  atmospheric  temperatures  reach  50° 
below  zero. 

*  Trans.  Am.  Soc.  C.E.,  Vol.  LXXV,  p.  219. 


18  HIGH    MASONRY    DAM    DESIGN 

"  (3)  For  storage  reservoirs  where  the  water  will  be 
drawn  off  each  year  during  the  winter  to  a  level  where 
the  dam  is  strong  enough  to  resist  the  ice  pressure. 

"  (4)  For  reservoirs  where  the  contour  of  the  ground 
at  the  high-water  level  is  such  that  the  expansive  force 
of  the  ice  will  not  reach  the  dam." 

Ice  thrust  has  a  greater  influence  on  the  thickness 
of  a  dam  than  the  flood  water  level,  down  to  about  no 
feet  below  the  flow  line  in  the  Olive  Bridge  Dam,  at 
which  point  the  lo-foot  flood  level  begins  to  require  a 
wider  base;  and  in  the  Kensico  Dam,  with  a  flood  level 
.of  5  feet,  the  change  is  about  210  feet  below,  indicating 
that,  for  dams  of  moderate  height,  ice  pressure  has  a 
very  great  influence.  (Cf.  5  Profiles  of  Fig.  i,  page  13.) 

In  the  Olive  Bridge  Dam  it  was  assumed,  that  clear 
block  ice  i  foot  thick  might  be  expected  to  form  at  the 
surface,  and  expand  so  as  to  exert  its  full  crushing  strength 
of  about  47,000  pounds  per  linear  foot  of  dam,  and  this 
figure  was  used  in  the  Wachusett  Dam;  42,000  pounds 
per  linear  foot  was  recommended  in  the  Quaker  Bridge 
Dam,  and  30,000  pounds  per  linear  foot  in  the  Croton 
Falls  Dam,  and  24,000  in  Cross  River  Dam,  while  in  the 
Design  of  the  New  Croton  Dam  ice  thrust  was  dis- 
regarded. 

In  a  discussion  regarding  ice  pressure,  before  the 
Canadian  Society  of  Civil  Engineers,  in  December,  1891, 
agreement  seemed  to  have  been  reached  on  two  points: 
That  thrust  from  ice  less  than  3  inches  thick  can  be  dis- 
regarded, and  that  the  thrust  can  safely  be  taken  at  the 
crushing  strength  of  ice.  The  ' '  Engineering  News  "  of  Jan- 
uary 12,  1893,  and  of  April  5,  1894,  records  the  compressive 


HIGH    MASONRY    DAM    DESIGN  19 

p  -  ; 

strength  of  ice  ranging  all  the^way  from  100  to  1000  pounds 
per  square  inch;  21,500  pounds  per  linear  foot  of  dam, 
with  ice  at  3  inches  thick  would  be  equivalent  to  about 
600  pounds  resistance  per  square  inch.  The  value  of  47,000 
pounds  above  noted  is  equivalent  to  about  650  pounds 
per  square  inch  for  ice  6  inches  thick. 

The  uncertainties  of  the  problem  are  thus  increased 
by  the  lack  of  exact  and  more  extended  data  bearing  upon 
the  subject  of  ice  pressures.  It  is,  therefore,  largely  a 
matter  of  judgment  as  to  what  should  be  used  as  an  ice 
thrust  in  any  given  case,  as  has  been  indicated  by  the 
foregoing  discussion. 


CHAPTER  II 
PRELIMINARY   CONSIDERATIONS 

THE  studies  involved  in  the  determination  of  a  gravity 
cross-section  demand  an  investigation  along  two  general 
lines : 

First,  the  direct  calculation  to  fix  the  most  economical 
cross-section  under  the  imposed  conditions,  and 

Second,  studies  in  the  comparison  of  cross-sections 
ranging  between  this  one,  which  may  be  called  the  min- 
imum, and  one  of  an  existing  masonry  dam,  where  the 
conditions  and  responsibility  are  practically  the  same  as 
those  under  consideration. 

Before  undertaking  such  an  analysis,  however,  it  will 
be  advisable  to  consider  the  manner  in  which  water  pressure 
is  exerted  against  a  submerged  surface;  its  amount;  the 
method  of  determining  the  point  of  s  application  of  the 
resultant ;  the  assumed  distribution  of  pressure  in  a  masonry 
joint ;  and  finally,  the  action  of  the  forces  in  and  upon  the 
structure. 

It  may  be  stated  as  a  general  proposition  that  water 
pressure  acts  in  all  directions  against  a  submerged  object 
and  that  it  depends  for.  its  value  merely  upon  the  "  head," 
or  depth  of  the  center  of  gravity  of  the  figure  below  the 
free  surface  of  the  liquid.  In  consequence  of  this  principle 
it  may  be  shown  that  the  total  normal  pressure  is  repre- 
sented by 

(i) 

20 


HIGH    MASONRY   DAM   DESIGN  21 

» 

where  P=the  total  normal  prdtesure; 

7-=  the  weight  of  a  unit  volume  of  water; 
A  =the  total  area  ;  and 

h=ihe   vertical   distance    of   the  figure's   center   of 
gravity  beneath  the  free  surface  of  water. 

The  demonstration  *  may  be  made  by  considering  the 
surface  to  be  divided  into  an  infinite  number  of  parts;  the 
total  pressure  on  each  one  of  these  elements,  depending 
only  upon  the  weight  of  water  resting  upon  it,  may  be 
written, 

.......     (2) 


in  which  p  =  the  total  normal  pressure  on  the  differential 

area; 

a  =  the  differential  area  ; 

hi  =  the  head  on  a  (practically  constant  over  the 
differential  area). 

If,  therefore,  we  take  the  sum  of  the  pressures  on  all  of 
these  small  areas,  we  shall  obtain  the  previous  equation, 
which  is  perfectly  general  and  applies  to  any  surface. 
In  the  case  of  a  vertical,  rectangular  strip  of  the  back 
of  a  dam,  the  application  of  the  formula  will  give  a  total 
pressure  of 


p--J*f 

Jo 


where  b  is  the  constant  breadth  of  the  strip,  usually  taken 
as  one  unit,  oc  is  the  variable,  and  H  is  the  total  height 
of  the  rectangle. 

*  See  Merriman's  "  Hydraulics." 


22  HIGH    MASONRY   DAM    DESIGN 

The  point  on  the  submerged  surface  at  which  this 
resultant  pressure  acts  may  be  determined  by  assuming 
for  an  axis  the  horizontal  line  in  which  the  surface  of 
the  water  cuts  the  plane  of  the  back  of  the  dam,  taking 
the  moment  of  inertia  of  the  surface  about  this  axis,  and 
dividing  the  result  by  the  static  moment  of  the  surface 
with  reference  to  the  same  axis.  Applying  this  to  the 
strip  referred  to  above,  there  will  result, 

bH3 
1= ,  the  moment  of  inertia  of  rectangular  strip, 

and 


12 


which  is  the  moment   of   inertia  with   regard   to  the  as- 
sumed axis. 

H     bH2 


is  the  static  moment  of  the  surface  about  the  same  axis, 
hence, 

/!         2H 


is  the  distance  of  the  center  of  pressure  from  the  surface 
of  the  water. 

In  the  investigation  of  the  distribution  of  pressure  in 
a  masonry  joint  subjected  to  external  forces,  the  material 
is  assumed  to  be  rigid,  though  in  reality  it  is  to  a  certain 


HIGH   MASONRY  DAM   DESIGN  23 

> 

degree  elastic.  This  elasticity  gives  the  distribution  of 
stress  an  indeterminate  law,  so  that  neither  the  direction 
nor  the  intensity  is  actually  known  at  any  point.  It  is 
certain,  however,  that  the  intensity  must  be  zero  at  the 
edges,  although  it  may  increase  with  great  rapidity  to 
higher  values  very  near  the  limits  of  the  joint.  Investi- 
gations have  been  made  within  the  past  few  years  to 
obtain  more  exact  information  as  to  this  distribution  of 
stress,  but  so  far  the  results  are  not  completely  satisfac- 
tory. Reference  will  be  made  to  this  matter  in  Chapter 
VIII. 

Inasmuch  as  the  exact  law  of  stress  variation  is  not 
known,  one  of  uniform  variation  of  normal  stress  has 
been  assumed  in  all  practical  treatments  of  masonry 
joints. 

Fig.  2  represents  the  simplest  case,  in  which  the 
pressure  is  assumed  to  be  uniformly  distributed  over  the 
joint  a  6,  with  the  constant  intensity  p\  it  might  be  taken 
as  representing  any  horizontal  joint  with  a  superimposed 
load  acting  at  its  center. 

To  express  this  condition  of  uniform  stress  algebra- 
ically, /  may  be  assumed  to  be  the  length  of  the  joint 
from  a  to  b,  while  the  breadth,  perpendicular  to  the  plane 
of  the  paper,  is  taken  as  unity.  The  area  of  the  joint 
will  then  be  /,  whence, 

W=pl,      .......     (5) 

or 

W 


which  is  the  formula  for  a  condition  of  uniform  intensity  of 
stress  over  the  entire  joint. 


A  W|  I 

U it- — -->{ 

m///////////////////////////W//////////////////////////tffo> 


24  HIGH   MASONRY    DAM   DESIGN 

It  may  be  observed  here  that  this  pressure  is  uniform 
only  because  the  total  load  represented  by  W,  acts  at 
the  center  of  the  joint,  and  that 
when  the  point  of  application  is 
changed  to  some  other  position,  a 
there  will  be  an  increased  stress 
in  that  direction  toward  which 
the  load  has  been  moved,  and 
a  corresponding  decrease  in  the 
opposite  direction. 

It  will  be  necessary  therefore,  to  consider  this  varia- 
tion of  pressure  in  eccentrically  loaded  joints  and  also 
the  manner  in  which  the  eccentricity  in  the  case  of  a  dam 
is  produced. 

If  a  b  be  any  plane,  horizontal  joint  in  the  dam  at  the 
distance  H  below  the  surface,  OY  the  water  surface,  and 
<j>  the  angle  that  the  back  makes  with  the  vertical,  then 
the  total  pressure  on  the  back  acting  at  a  point  one- third 
the  distance  up  from  the  joint,  will  be 


sec 


Combining  this  force  with  the  weight  of  masonry  W 
above  the  joint  acting  through  the  center  of  gravity  of  the 
section,  the  resultant  R  will  intersect  it  at  some  point  as 
e,  on  a  b,  other  than  the  center  of  figure,  called  the  center 
of  resistance,  and  it  is  evident  that  with  a  variation  of 
F'  and  W  it  may  occupy  any  position  along  the  joint. 

Fig.  3,  showing  only  the  vertical  component,  exhibits 
such  a  case,  where  compression  exists  over  the  entire  joint 


wi 


L — -u---  .>•) 

'A! 


HIGH   MASONRY  DAM  DESIGN  25 

as  in  Fig.  2,  but  where  the -center  of   pressure  is  not  at 
the  center  of  figure 

If  the  intensity  of  pressure 
at  b  may  be  represented  by 
the  vertical  line  p,  and  the  in- 
tensity of  pressure  at  a  by 
the  line  p',  then,  since  by  the 
assumption  the  pressure  varies 
uniformly  over  the  entire  joint, 
the  vertical,  p",  at  any  point,  included  between  the 
horizontal  a  b  and  the  line  joining  the  extremities  of  p 
and  pf  will  indicate  the  intensity  of  pressure  at  that 
point,  while  the  area  of  the  trapezoid  will  represent  the 
total  pressure  on  the  joint. 

The  former  may  be  expressed  algebraically  thus: 


FIG.  3. 


and  the  latter  by, 


I, 


(7) 


(8) 


The  determination  of  the  maximum  and  minimum 
pressure  p  and  pf  may  be  made  as  follows : 

Since  the  static  moment  of  the  rectangle  p  I  about  a 
point  J-/  from  p'  is  the  same  as  the  static  moment  of  the 
trapezoid  about  the  same  point,  because  the  moment  of 
the  triangle  p— p',  I  about  that  point  is  zero,  that  being 
the  center  of  gravity  of  the  triangle,  there  will  result  by 
taking  moments 


(9) 


26  HIGH  MASONRY  DAM  DESIGN 

whence, 

aW 


which  is  an  expression  for  the  intensity  of  pressure  at  the 
point  b,  on  the  joint  a  b.  To  solve  for  the  value  of  p'9 
the  intensity  of  the  pressure  at  the  point  a,  in  a  similar 
manner  we  may  take  moments  about  a  point  JZ  from  b, 
whence, 


or, 

.      ,       .       .        (12) 

When  pf  becomes  zero,  the  trapezoid  reduces  to  a  tri- 
angle as  shown  in  Fig.  4,  with  its  center  of  gravity  at  a 
distance  from  b  equal  to  f /,  and,  since  the  center  of  pres- 
sure of  W  must  lie  vertically  above  the  center  of  gravity 
of  the  triangle  graphically  representing  the  variation  of 
pressure  over  the  joint,  we  shall  have,  p'  =  o>  «  =  £/,  and 
Eq.  (9)  reducing  to 

Wl  =  pP. 
whence, 

2_W/ 


That  is  to  say,  the  maximum  pressure  p  for  this  condition 
is  twice  the  value  as  obtained  from  Eq.  (6), 


HIGH    MASONRY    DAM   DESIGN 


27 


In  Fig.  5  is  represented  &  case  in  which  tension  exists 
over  a  portion  of  the  joint,    p'  is  here  negative. 


Wl 


FIG,  4. 


FIG.  5. 


Although  both  masonry  and  the  best  hydraulic  cement 
mortar  have  considerable  tensile  strength,  running  up  to 
several  hundred  pounds  per  square  inch  in  tests,  the 
latter,  together  with  the  continued  adhesion  of  the  mortar 
to  the  aggregate  in  concrete,  when  used,  is  of  uncertain 
value  in  this  connection.  The  tensile  strength  is  therefore 
always  neglected  in  considering  the  stability  of  masonry 
dams  or  other  similar  structures,  and  is  an  omission  which 
is  the  more  justifiable  since  it  leads  to  an  error  on  the 
side  of  safety. 

In  the  case  represented  by  Fig.  5,  the  triangle,  whose 
base  is  3*4,  and  altitude  p,  is  therefore  alone  considered, 
and  by  taking  moments  about  6,  there  will  result, 


whence, 


Wu  =  3M  —  u  ,. 


2W 


(15) 


(16) 


If  it  is  desirable  to  know  what  the  tension  in  the  joint 


28  HIGH   MASONRY   DAM    DESIGN 

is,  it  may  be  determined  from  Eq.  (12"*.    As  ^<  i.o,  the 

resulting  value  is  negative,  thus  denoting  a  tension  by 
that  equation. 

The  pressures  at  a  and  b  may  also  be  determined  as 
follows:  Decomposing  the  resultant  acting  on  any  joint 
into  its  vertical  and  horizontal  components,  V  will  repre- 
sent the  total  normal  or  vertical  pressure,  equal  to  W, 
the  weight  of  masonry  above  the  joint,  plus  the  vertical 
component  of  the  thrust  from  the  water.  The  horizontal 
component  of  the  resultant  is  disregarded,  as  its  effect 
upon  the  joint  is  more  or  less  indeterminate,  and  since 
too,  it  is  assumed  to  be  neutralized  by  the  friction  acting 
in  the  joint. 

The  vertical  component  V,  acting  through  the  point  of 
application  of  the  resultant  R  in  the  joint,  is  therefore 
the  factor  producing  the  difference  in  pressure  between 
a  and  b,  or  the  uniformly  varying  stress. 

Assume  that  at  the  center  of  the  joint,  which  is  not 
necessarily  vertically  below  the  center  of  gravity  of  the 
mass  above,  two  forces  equal  and  opposite  to  each  other, 
and  of  the  same  value  V,  are  applied  normal  to  the  joint. 
The  effect  of  each  is  to  neutralize  the  other,  but  if  we 
consider,  apart  from  the  other  forces,  the  one  acting  down- 
ward, since  it  is  applied  at  the  center  of  figure  it  will 

V 
produce  a  uniform  stress  p  over  the  joint  equal  to  y. 

The  two  remaining  and  equal  forces  V  and  V,  one 
acting  downward  at  the  point  of  application  of  R,  and  the 
other  upward  at  the  center,  form  a  couple  whose  lever 
arm  is  v,  and  the  moment  of  which  is  therefore 


HIGH  MASONRY   DAM   DESIGN  29 


This  moment  produces  a  ufliformly  varying  stress  over 
the  joint,  increasing  the  intensity  at  b  and  decreasing  it 
at  a  by  an  equal  amount. 

To  determine  its  value  we  have  but  to  consider  the  fol- 
lowing : 

M  =  Vv.        ......     (17) 

the  moment  caused  by  the  couple  and  producing  the 
varying  stress.  Also, 


(18) 


where  k  is  the  intensity  of  stress  at  the  maximum  dis- 
tance from  the  neutral  axis;  7,  the  moment  of  inertia 
of  the  section  about  such  an  axis;  and  d\  the  normal  dis- 
tance from  the  neutral  axis  to  that  point  where  k  exists. 
Since  the  neutral  axis  passes  through  the  center  of 
figure  of  the  joint,  the  value  of  di  is  half  the  length  of  the 
joint,  while  /,  the  moment  of  inertia,  equals  TV3,  if  we 
consider  a  horizontal  section  in  the  plane  of  the  joint  a  b 
extending  back  from  the  plane  of  the  paper  one  unit's 
distance.  Hence, 

...  kl     fc3     W 


or, 

6  Vv 


Here  k  represents  the  stress  that  must  be  added  to  the 
uniform  stress        to  find  the  intensity  of  pressure  at  the 


30  HIGH   MASONRY   DAM    DESIGN. 

y 
toe  b  and  the  amount  which  must  be  subtracted  from  -y 

to  arrive  at  the  intensity  at  the  heel  a.  It  is  expressed  in 
pounds  per  square  inch,  but  if  the  distances  are  measured 
in  feet  and  the  forces  in  pounds,  k  will  be  designated  in 
pounds  per  square  foot. 

While  it  is  customary  to  consider  only  the  normal 
component  of  the  resultant  pressure  acting  in  a  hori- 
zontal joint  and  to  assume  it  to  vary  uniformly,  this  is 
probably  correct  only  for  horizontal  joints  in  rectangular 
walls  vertically  loaded  and  not  subjected  to  lateral  pres- 
sures. It  will  be  shown  later  that  the  maximum  stresses 
exist  at  or  near  the  down-stream  face,  and  act  in  a  direction 
parallel  to  and  on  planes  normal  to  that  face.  The  fact 
also  that  acute  edges  do  not  crack  off  in  the  inclined  faces 
of  dams  is  in  itself  a  partial  confirmation'  of  the  statement. 

Under  these  circumstances  then,  the  maximum  normal 
pressure  in  a  horizontal  joint  must  be  much  less  than  the 
actual  maximum  pressure  in  the  dam,  and  it  has  been 
assumed  to  bear  the  ratio  to  the  latter  of  about  9  to  13. 


I 
I 


CHAPTER  III— PART  I 
DEVELOPMENT  OF  FORMULAE  FOR   DESIGN 

Six  series  of  formulae,  designated  by  the  letters  A,  B, 
C,  D,  E,  and  F,  will  now  be  presented,  in  each  of  which 
a  given  set  of  conditions  with  respect  to  the  external 
forces  will  be  involved;  but  as  the  method  of  procedure 
is  practically  the  same  for  all  cases,  only  series  A  will 
be  developed  here. 

The  following  nomenclature  will  be  employed: 

L  =  the  width  of  the  top  of  the  dam  cross-section; 
/  =  length   of   a  horizontal  joint   of  masonry,   to 

be  determined; 
10  =  known  length  of  the  joint  next  above  joint  of 

length  /; 
h  =  depth  of  a  course  of  masonry  (vertical  distance 

between  /0  and  /) ; 
P=line  of  pressure,  reservoir  full; 
Pf  =  line  of  pressure,  reservoir  empty ; 
u  =  distance  from  front  edge  of  the  joint  /  to  the 

point  of  intersection  of  P  with  the  joint  /, 

measured  parallel  to  joint  /; 
y  =  distance  from  back  edge  of  the  joint  /  to  the 

point  of  intersection  of  P'  with  the  joint  /. 

measured  parallel  to  joint  /; 
y0  =  distance   from    back   edge  of    the  joint  /0  to 

the   point   of    intersection    of   Pf   with    the 

joint  /0,  measured  parallel  to  joint  1Q\ 

31 


32  HIGH    MASONRY    DAM    DESIGN 

v  =  distance   between   P   and  P'   at   the  joint   /, 

measured  parallel  to  joint  /; 
?•  =  weight  in  pounds  of  a  cubic  foot  of  water 

(62.5); 
f  =  weight   in   pounds   of   a   cubic   foot   of   mud 

(75-90); 
J  =  ratio  of  unit  weight  of  masonry  to  unit  weight 

of  water   (often  assumed  as  J) ; 
J^-  =  wTeight  in  pounds  of  a  cubic  foot  of  masonry; 
//=head  of  water  on  joint  /  (vertical  distance  of 

joint  /  below  water  surface) ; 

H' =  depth  of  earth  back  fill  over  joint  /  on  front; 
HI  =head  of  water  on  joint  /  when  ice  acts  at  sur- 
face of  water; 
H  —  Hi=rise  of  water  level,  due  to  flood,  wave,  etc., 

above  normal  level  for  full  reservoir; 
/*i=head  of  water  above  mud  level   (liquid  mud 

of  weight  f) ; 

/J2=head  of  liquid  mud  on  joint  /,  on  back; 
a  =  vertical  distance  from  the  top  of  the  dam  to 

the  surface  of  water  (flood) ; 

ai  =  vertical  distance  from  the  top  of  the  dam  to 
the  surface  of  water  when  ice  is  considered 
(ai  generally  exceeds  a) ; 
6  =  vertical   distance   from   water  surface   to   top 

of  dam  when  dam  is  overtopped; 
c  =  ratio    of    upward    thrust    intensity,    due    to 
hydrostatic  head  H  (or  HI,  or  hi+h2),  as- 
sumed  to   act   at   heel   of  joint   /    (usually 
assumed  as 


HIGH    MASONRY    DAM    DESIGN  33 

> 

Tf= horizontal  ice  thrust  at  water  surface  in  pounds 

(47,000); 

(The  value  here  given,  for  example,  was  used 
in  studies  for  design.  Our  present  lack 
of  exact  data  in  regard  to  ice  pressures 
prevents  more  than  a  speculation  from 
being  made  as  to  a  definite  value  to  be 
assigned  in  any  case); 

Df  =  horizontal  dynamic  thrust  of  water  in  pounds;* 

Ef  =  thrust  of  earth  back  fill  in  pounds  (on  front); 

Wvf  =  vertical   pressure   on    inclined   upstream   face 

above  joint  I,  in  pounds; 
AQ  =  total    area    of    cross-section    of    dam    above 

joint  /0; 
A  =  total    area    of    cross-section    of    dam    above 

joint  /. 

/  =  batter  of  upstream  face  for  vertical  distance  h ; 
5  =  distance  of  line  of  action  of  Wvf  from  upstream 
edge  of  joint  /,  measured  parallel  to  joint  /; 
d  =  angle  that  Ey  makes   with  horizontal; 
a.  =  angle   of   slope   of   downstream   face   of   dam 

with  horizontal; 

/?  =  angle  R  makes  with  the  vertical; 
p  =  maximum  allowable  pressure  intensity  at  toe 

(in  pounds  per  square  foot) ; 

q  =  maximum  allowable  pressure  intensity  at  heel 
(in  pounds  per  square  foot)  (p  is  assumed 
less  than  q)  p  and  q  may  be  used  to  signify 
the  calculated,  existent  pressure  intensities 

*  Determined,  as  in  the  case  of  the  overfall  dam,  by  the  probable  velocity 
of  flow  against  the  dam. 


34  HIGH   MASONRY   DAM   DESIGN 

corresponding  to  P  and  P'  respectively,  for 

the  joint  /. 
/=the    coefficient    of    friction    for    masonry    on 

masonry  (usually  0.6  to  0.75); 
5  =  the   shearing   resistance   of   the  masonry   per 

square  unit; 

rH2 

F  =  ~ — =the  horizontal  static  thrust  of  the  water  in 
2 

pounds ; 

the   moment   of  F  about   any   point   in   the 

joint  /; 

total    weight,    in    pounds,    of    masonry 
resting  on  the  joint  /; 

total    weight,    in    pounds,    of    masonry 
resting  on  the  joint  /0; 
R  =  the  resultant  of  F  and  W\ 
Rr  =  the  resultant  of  the  reactions ; 

—  =  up  ward  thrust  of  water  on  base  /. 

In  the  figures,  hydrostatic  pressures  are  indicated  by 
triangular  and  trapezoidal  areas  included  within  dotted 
lines,  while  ice  pressure  is  shown  to  contrast  HI  with  H. 

As  before,  if  a  unit  length  of  one  foot  of  dam  be 
considered,  the  letters  T,  D,  E,  W9,  A,  A0,  and  H2  will 
signify  volumes. 

It  will  be  observed  that,  where  possible,  the  several 
equations  have  been  cleared  of  the  term  Ar,  thereby 
simplifying  actual  calculations. 

In  the  above  table  c,  in  a  manner,  may  be  considered 
to  provide  for  an  assumption  of  a  certain  proportion  of  the 


HIGH    MASONRY    DAM    DESIGN  35 


..; 


joint's  area  being  subjected  to  upward  water  pressure ;  and 
the  distribution,  as  evidenced  by  cHlf/2,  varying  from 
a  maximum  intensity  at  the  heel  to  zero  intensity  at  the 
toe,  is  assumed  in  view  of  the  fact  that  the  tendency  to 
open  the  joint  would  begin  at  the  heel  while  a  zero  intensity 
of  upward  pressure  at  the  toe  would  presuppose  an  opening 
with  consequent  flow  at  that  point.  As  the  dam  would 
then  be  failing  in  its  chief  function,  i.e.,  to  retain  water, 
this  flow  is  not  considered  to  exceed  a  slight  seepage. 

In  general  four  ways  are  recognized  in  which  a  masonry 
dam  may  fail : 

1.  By  overturning  about  the  edge  of  any  joint,   due 
to  the  line  of  action  of  the  resultant  passing  beyond  the 
limits  of  stability. 

2.  By    the    crushing    of    the    masonry    or    foundation 
because  of  excessive  pressure. 

3.  By  the  shearing  or  sliding  on  the  foundation  or  any 
joint,  due  to  the  horizontal  thrust  exceeding  the  shearing 
and  frictional  stability  of  the  material. 

4.  By  the  rupture  of  any  joint  due  to  tension  in  it. 
An  unsatisfactory  foundation  might  also  be  mentioned 

as  possibly  leading  to  failure,  and  in  view  of  this,  the 
footing  upon  which  the  dam  rests  should  always  be  most 
carefully  scrutinized. 

To  preclude  failure  from  any  of  the  above  mentioned 
causes,  it  is  the  practice  to  design  the  cross-section  of 
the  dam  with  the  following  conditions  imposed : 

1.  The   lines   of   pressure,    both   for   the   reservoir   full 
and  empty,  must  not  pass  outside  the  middle  third  of  any 
horizontal  joint. 

2.  The    maximum    normal    working    pressure    on    any 


36  HIGH    MASONRY  DAM    DESIGN 

horizontal  joint  must  never  exceed  certain  prescribed 
limits,  either  in  the  masonry  itself  or  in  the  founda- 
tion. 

3.  The  coefficient  of  friction  in  any  plane  horizontal 
joint,  or  between  the  dam  and  its  foundation,  must  not  be 
less  than  the  tangent  of  the  angle  which  the  resultant 
makes  with  a  vertical. 

As  may  be  seen  by  referring  to  the  figures  showing 
the  distribution  of  pressure  on  a  joint,  when  the  resultant 
lies  within  the  middle  third,  tension  can  exist  in  no  part 
of  it,  nor  can  the  safety  factor  be  less  than  two,  if  we 
neglect  to  consider  the  upward  pressure  of  water  perco- 
lating through  any  of  the  joints  or  beneath  the  dam. 


U y ><- — v >r< u 


FIG.  6. 

To  illustrate  the  conditions  that  exist  and  to  derive 
the  value  of  the  safety  factor  when  the  resultant  cuts  the 
joint  at  the  extremity  of  the  middle  third,  we  may  take 
the  case  as  shown  in  Fig.  6.  Resolving  R  into  its  horizontal 
and  vertical  components,  and  taking  moments  about  the 
center  of  resistance  e,  the  following  equation  is  obtained: 


where  F  is  the  horizontal  component  of  the  thrust  from 
the  water  behind  the  dam,  acting  at  a  point  \H  above 


HIGH    MASONRY   DAM   DESIGN  37 

i 

the  plane  of  the  joint,  whij£  W  is  the  vertical  compo- 
nent of  the  resultant,  and  as  such,  includes  not  only 
the  weight  of  the  masonry,  but  the  vertical  component 
of  the  thrust  from  the  water  as  well,  provided  the 
latter  is  considered  as  acting  normal  to  the  back  of  the 
dam. 

For  the  dam  to  be  on  the  point  of  rotating  about  6, 
the  downstream  edge  of  the  joint,  it  is  obvious  that  the 
resultant  R  must  pass  through  that  point.  Under  these 

TT 

circumstances,  since  the  lever  arm  of  F  is  still  — ,  and  the 

o 

lever  arm  of  W  has  been  increased  to  twice  its  former 
value  or  2  ( — j ,  for  the  above  equation  to  still  hold,  F  must 

also  be  increased  to  twice  its  former  value.  This  would 
indicate  that  when  R  acts  through  the  point  e,  the  value 
of  H  is  only  one-half  as  great  as  is  necessary  to  produce 
overturning;  or,  in  other  words,  that  the  factor  of  safety 

is  two  as  indicated  by  the  ratio  of  —    — .     It  should  be 

observed  however,  that  the  material  near  the  edge  of  the 
joint  will  crush  some  time  before  the  resultant  has  reached 
it,  and  that  therefore  the  factor  of  safety  against  overturning 
with  R  at  the  limit  of  the  middle  third  is  something  less 
than  two. 

When,  however,  the  upward  pressure  of  water  acting 
over  the  joint  due  to  percolation  is  taken  into  considera- 
tion, the  factor  of  safety  will  be  somewhat  modified,  as  the 
following  demonstration  will  make  clear. 

By  referring  to  Fig.  7,  it  will  be  seen  that,  for  example, 

the  horizontal  water  pressure  on  the  back,  - — ,  the  uplift, 


38 


HIGH  MASONRY  DAM  DESIGN 


,  and  the  weight  of  the  masonry,  W,  constitute  the 


forces,  with  the  reaction  (not  shown)  considered  as  acting 
on  the  dam. 

The  forces  tending  to  overturn  the  dam  about  the 


toe  at  b  are  - —  and  — — ,  the  resultant  of  which  is  denoted 
2  2 

in  magnitude  and  direction  by  the  line,  0,  at  the  normal 
distance,  r,  from  b.  The  force  resisting  this  tendency  is 
W,  the  line  of  action  of  which  is  normally  distant  (u+v) 
from  b.  If  nm  denotes  the  force,  0,  in  magnitude  and 
direction,  and  n  dy  the  force,  W,  then  n  g  will  represent, 
in  magnitude  and  direction,  the  force,  R,  or  the  final  re- 
sultant of  all  the  forces,  which  is  opposed  by  the  reaction 


HIGH  MASONRY  DAM   DESIGN  39 

(for  equilibrium  to  be  assumed)  at  the  point,  e,  distant 
u  from  b. 

The  resisting  moment  about  &,  then,  is  W(u+v)=MQ 
while  the  overturning  moment  about  the  same  point  is 


Whence,  for  the  ratio  of  "  resisting  moment  to  overturning 
moment  "  there  may  be  written: 


MQ  MQ  (          . 

' 


and  for  the  ratio  of  the  "  resultant  moment  of  the  ver- 
tical components  "  to  the  "  resultant  moment  of  the 
horizontal  components  "  of  the  forces  there  follows: 


These  two  expressions  for  the  "  factor "  evidently 
become  equal  to  each  other  only  when  Mi  =  o,  or  when 
uplift  is  ignored;  also,  when  the  factor  of  safety  is  equal 
to  unity  (when  MQ— M2=Mi),  or  at  the  point  of  over- 

AM    — I— 7t 

turning.     Therefore,  the  usual  expression,  ,  will  not 

v 

be  the  value  for  the   "  factor  of  safety  "  against  over- 
turning, according  to  Eq.  (210). 

To  consider  the  foregoing  discussion  for  the  purpose  of 
developing  a  graphic  treatment  for  determining  the  value 
in  either  case,  it  must  be  remembered  that,  for  the  dam  to 


40  HIGH    MASONRY    DAM    DESIGN 

be  on  the  point  of  overturning,  the  ratio  of  the  two  moments, 
MQ  and  M'  of  Eq.  (2ia),  must  equal  unity  or  M0  =  M\  -\-Mi, 
and  the  line  of  action,  R,  in  Fig.  7,  must  pass  through  b. 
Inasmuch  as  W  is  constant,  one  or  all  of  the  other  forces 
may  at  this  stage  be  considered  variable,  in  order  to  bring 
about  the  above  supposititious  condition.  According  to 
Eq.  (2ia),  the  distance,  r,  is  constant,  and  the  water  pressure 
on  the  back  and  the  uplift,  therefore,  are  supposed  to  be 
proportionately  increased  to  fulfill  this  condition,  of  bring- 
ing the  line  of  action,  R,  through  b.  This  seems  reasonable 
from  the  fact  that  the  horizontal  water  thrust  cannot  be 
considered  to  increase  without  a  corresponding  increase 
in  the  uplift.  Therefore,  the  condition  necessary  to  bring, 
the  resultant,  R,  through  b  instead  of  through  e,  where 
it  actually  falls,  is  that  0=dg  be  increased  to  di,  in  Fig.  7. 
If  be  be  drawn  through  b,  parallel  to  0  (and,  therefore, 
to  dg),  the  ratio  sought  follows  from  the  similarity  of  the 
triangles,  nid  and  nbc,  or,  the  factor  of  safety,  with  respect 
to  resisting  moment  and  overturning  moment,  is  equal  to 

,.'    di     cb 

the  ratio,  —  =  -p. 

dg     cj 

The  foregoing  conception,  Eq.  (216),  of  the  "  factor  of 
safety  "  tacitly  assumes  that  only  the  horizontal  thrust 
of  the  water  is  instrumental  in  moving  the  center  of  pres- 
sure from  e  to  b,  and  that  the  uplift  merely  lessens  the 
resisting  moment. 

As  the  overturning  force  to  be  increased  is  therefore 
horizontal,  and  as  the  length  of  the  line  parallel  to  the 
overturning  resultant  and  comprehended  between  the 
point,  b,  and  the  line  of  action  of  the  resisting  force  is 
divided  by  its  segment  (comprehended  between  the  actual 


HIGH    MASONRY    DAM    DESIGN  41 

resultant  and  the  same  line  'of  action  of  the  resisting  force) 

•»< 
to  get  the  ratio,  or  factor  against  overturning,  it  at  once 

follows  that  the  division  according  to  Eq.  (216)  would  be 
—  .     Eq.  (210)  seems  preferable,  or  the  "  factor  of  safety" 

=  c—,  as  in  Fig.   7.     The  "factor  of  safety,"  however,  is 

CJ 
of  doubtful  value,  due  to  the  certain  impossibility  of  the 

structure's  rotation  about  the  point  6;  but  it  may  be  a 
useful  quantity  for  comparison  at  times.  The  expression 
(210)  may  give  values  less  than  (216)  by  as  much  as  one- 
third,  in  some  cases. 

As  was  stated  previously,  the  frictional  and  shearing 
resistance  of  a  joint  is  assumed  to  withstand  the  tendency 
of  the  horizontal  thrust  to  slide  the  upper  portion  over 
the  lower,  so  that  it  is  quite  customary,  even  though  it 
should  be  investigated,  to  neglect  it. 

For  equilibrium  in  this  regard, 


(22) 


where  F  is  the  horizontal  component  of  the  water's  thrust,  / 
the  coefficient  of  friction,  usually  taken  between  0.6  and 
0.75  for  masonry,  and  5  is  the  shearing  resistance  per  unit 
of  area. 

In  spite  of  the  fact  that  5  has  an  appreciable  value, 
and  particularly  so  for  monolithic  masses  of  "  cyclopean 
masonry,"  the  value  is  practically  indeterminate,  and 
consequently  usually  ignored.  Numerous  attempts  have 
been  made  however,  to  write  expressions  for  it,  the  most 
rational  of  which  depends  upon  the  trapezoidal  law  of 
the  distribution  of  normal  stress  ;  but  this  too  is  unsatis- 


42  HIGH    MASONRY    DAM    DESIGN 

factory  from  a  practical  standpoint.*     We  shall  neglect 
5,  therefore,  in  the  previous  equation,  whence, 


.     .    ,     .  ..•.-  (23) 
which  gives  at  the  limit, 

p 


(24) 


In  every  design  the  imposed  conditions  for  equilibrium 
result  in  a  cross-section  in  which  the  back  has  very  much 
less  of  a  batter  than  the  front.  It  may  be  shown  also,* 
that,  as  the  shear  along  either  face  is  zero,  the  greatest 
intensity  of  stress  will  act  in  a  direction  parallel  to  the 
face  at,  and  near,  the  edge.  Since  the  horizontal  compo- 
nent of  the  pressure  is  ignored,  this  implies  that  the 
greatest  vertical,  or  normal  working  intensity  of  pres- 
sure must  be  less  at  the  downstream  face  where  the 
inclination  is  greater  than  at  the  heel,  in  order  that  the 
components  parallel  to  the  respective  faces  shall  be  ap- 
proximately equal.  This  is  accomplished  by  using  a  smaller 
vertical  normal  working  stress  at  the  toe  than  at  the  heel. 

As  the  up-stream  face  of  a  masonry  dam  is  vertical  for 
a  considerable  distance  from  the  top,  and  then  becomes 
only  slightly  inclined  to  it,  it  is  customary  to  consider  the 
thrust  from  the  water  as  acting  horizontally.  This  is  the 
more  justifiable  since  the  vertical  component  of  the  water 
resting  upon  the  up-stream  face  of  the  dam  causes  an 
overturning  moment  about  the  center  of  resistance,  op- 
posite in  direction  to  that  induced  by  the  horizontal 
thrust,  and  hence  is  an  error  on  the  side  of  safety. 
*  See  Chapter  VIII. 


HIGH    MASONRY   DAM    DESIGN  43 

> 

It  must  be  evident  fron*  the  equation  of  pressure, 
p  =  -pah,  that  where  this  alone  governs  the  resulting  theo- 
retical cross-section,  it  will  be  triangular  in  form  with  the 
apex  at  the  surface  of  the  water;  but  where  it  is  intended 
there  shall  be  no  flow  over  the  crest  of  the  dam,  it  is 
customary  to  carry  the  masonry  some  distance  above 
the  elevation  of  the  water  in  the  reservoir,  not  only  to 
allow  for  fluctuations,  but  because  of  economic  condi- 
tions or  to  provide  for  a  foot  or  carriage  way.  The  super- 
elevation and  the  width  of  top  are  therefore  arbitrarily 
assumed  and  should  be  taken  at  about  TV  the  height 
of  the  dam,  with  a  minimum  width  of  5  feet  and  a  maxi- 
mum superelevation  of  20  feet.* 

As  no  equation  can  be  written  simultaneously  ex- 
pressing the  three  conditions  of  stability,  i.e.,  that  the 
resultant  lie  within  the  middle  third,  that  the  maximum 
pressures  shall  not  exceed  certain  limits,  and  that  the 
horizontal  components  shall  not  cause  sliding,  it  be- 
comes necessary  to  determine  the  length  of  joints  (usually 
taken  vertically  10  feet  apart  for  a  depth  of  about  100 
feet  and  increasing  to  20  or  30  feet  below),  by  the  aid  of 
that  equation  involving  the  limiting  conditions  which  are 
known  to  apply,  in  order  that  the  cross-section  be  a  min- 
imum, and  then  to  test  the  joint,  if  necessary,  by  the 
other  two.  Generally  speaking  the  third  condition  will 
be  found  to  hold  if  the  joint  has  been  designed  in  accord- 
ance with  the  other  two. 

*Mr.  William  P.  Creager,  in  Proc.  Am.  Soc.  of  C.E.,  for  Nov.,  1915, 
"  The  Economical  Top  Width  of  Non-overflow  Dams,"  shows  this  width  to 
lie  between  10%  and  17%  of  the  height,  according  to  design  assumptions  and 
concludes  that  exceptionally  wide  tops  may  be  used,  there  being  but  slight 
economy  in  adopting  narrow  tops. 


44  HIGH    MASONRY    DAM    DESIGN 

Considering  Fig.  6,  in  which  /  is  the  length  of  joint, 
it  is  seen  to  be  divided  .  into  three  parts,  u,  v,  and  y, 
and  from  this  what  may  be  called  the  fundamental  equa- 
tion of  the  entire  design  can  be  written. 


(25) 


If  M  represents  the  overturning  moment  about  e,  then 
we  have  that  at  the  limit  of  the  middle  third, 

M  =  F-  =Wv,      .     .     ,     .     .     (26) 

or, 

M  . 

v=w.         .     .  ~;     .     -   .     .     (27) 

As  the  analysis  will  result  in  a  cross-section  polygonal 
in  outline,  composed  of  trapezoids  with  bases  /  and  1Q 
and  altitudes  h,  we  may  write  a  general  equation, 


.     .     .          (28) 
or, 


whence, 
\.    /  '  A-A0+(^)k,        .....     (29) 

and  since; 

W  fl  +  l°\j 

Tr=A»+(—)h> 
then, 

M_ 

f     x 
.     ,     .     .     .     (30) 


HIGH    MASONRY    DAM    DESIGN  45 


which  value  of  v,  if  substitute^  in  Eq.  (25)  gives, 


M_ 
Ar 


The  above  Eq.  (31)  is  a  modification  of  Eq.  (25)  and, 
when  proper  values  have  been  assigned  to  u  and  y,  depend- 
ing upon  the  existing  conditions,  is  used  throughout  the 
entire  design  in  the  determination  of  the  length  of  joints. 

In  the  upper  rectangular  portion  of  the  dam,  where 
there  is  an  excess  of  material  above  that  required  by  the 
static  pressure  of  the  water,  it  will  be  found  unnecessary 
to  consider  failure  from  crushing,  as  the  maximum  normal 
pressures  are  well  below  the  allowed  working  pressure, 
and  consequently  the  depth  at  which  the  section  ceases 
to  be  rectangular  will  be  fixed  by  the  fact  that  the  re- 
sultant may  not  pass  outside  the  middle  third.  The 
algebraic  expression  for  this  condition  is, 

u  ^  J  /  for  reservoir  full,        .    •.     .     .     (32) 
y  ^  J  /  for  reservoir  empty.        .     .     .     (33) 

Below  this  rectangular  portion,  trapezoidal  sections 
will  be  found.  At  the  base  of  the  rectangle,  l  =  l(\=L, 

u  =  —,   and,   since  the  center  of  gravity  of  the  figure  is 
o 

vertically  above  the  center  of  the  joint,  y  =  —. 

If  we  wish  to  determine  the  depth  to  which  the  rectan- 
gular portion  extends,  we  may  do  so  by  the  use  of  Eq. 


46  HIGH    MASONRY    DAM    DESIGN 

(31),   which,  as  shown,  must  involve  the  condition  that 
the   resultant   shall   just   touch    the   limit    of   the   middle 

third,  i.e.,  u=  —  .     Substituting  in  Eq.    (31),  u  =  —  ,  y  =  —, 
and  remembering  that  A0  =  o 


_L          64          L 
- 


whence,  by  dividing  by  L, 


or,  solving  for  H,  _ 

.....     -(34) 


If  H=h  then  a=0,  and  Eq.  (34)  reduces  to, 

.     .....     (35) 

At  this  depth  the  rectangle  ceases,  the  sections  become 
trapezoidal,  the  back  face  is 
still  vertical  but  the  front 
face  hf,  is  inclined  in  order 
to  increase  the  length  of  the 
successive  joints  and  thus 
maintain  the  resultant  for 
the  reservoir  full  at  the  down- 
stream limit  of  the  middle 
third.  For  a  considerable  distance  below  the  rectangular 

section  therefore,  Eq.  (31)  will  be  used  with  u  =  --  to  de- 

6 


HIGH    MASONRY    DAM    DESIGN  47 


t  ermine  the  length  of  Joint,  afod  the  back  face  will  remain 
vertical,  but  for  each  new  joint  the  resultant  for  the  reser- 
voir empty  will  approach  nearer  and  nearer  to  the  limits 

of  the  middle  third,  until  finally  y=—. 

O 

It  is  therefore  expedient  to  compute  the  value  of  y 
under  these  conditions  to  learn  exactly  at  what  vertical 

depth  or  joint  this  value  of  y  first  equals  -. 

O 

To  do  this,  moments  are  taken  about  the  vertical 
face,  for  both  A0  and  the  trapezoid,  the  latter  being  found 
by  dividing  the  trapezoid  into  a  rectangle  and  a  triangle; 
its  value  is, 


and  hence, 


Substituting  the  value  of  A  from  Eq.  (29)  in  the  above, 
and  solving  for  y  there  results, 


•   •   •   (36) 


.4, 


This  gives  a  value  of  y  to  be  substituted  in  Eq.   (31) 
while    the   value   of   u  =  J/,    which   has    been    maintained 


48  HIGH    MASONRY    DAM    DESIGN 

since  leaving  the  bottom  of  the  rectangular  section,   is 
substituted  also.     There  then  results  by  reduction, 


»,  •  •  (37) 


which  is  the  equation  used  in  the  determination  of  the 
length  of  joint  from  the  foot  of  the  rectangular  section 
down  to  that  joint  where  Eq.  (36)  first  gives  a  value  of 

y=-.     At  this  point  the  back  face  must  be  made  to  slope, 

o 

while  w=y  =  \l  is  substituted  in  Eq.  (31)  to  obtain  the 
following: 


which  will  determine  the  length  of  the  joints. 

The  second  condition  will  be  a  factor  from  here  on, 
for  below  this  section  at  some  point,  the  intensities  of  the 
pressures  at  the  toe  will  gradually  approach  and  finally 
equal  the  allowable  limit  ^  and  the  length  of  the  joint 
will  depend  primarily  upon  this.  It  is  therefore  necessary, 
after  each  application  of  Eq.  (38)  to  see  if  the  limiting 
pressure  p  at  the  toe,  which  is  smaller  than  q,  at  the  heel, 
has  been  reached.  Its  value  is  derived  from  the  equation 

2W     zAAr  . 

p  =  —j-  = — 7— -,  and  when  the  limiting  value  of  p  has  been 

realized  the  value  of  u  thereafter  must  be  derived  from, 


.       . 

(39) 


HIGH    MASONRY    DAM    DESIGN  49 

j& 

in  which  u,  is  seen  to  be  dependent  upon  the  normal  working 
pressure  p  at  the  toe.  (Eq.  (39)  follows  directly  from 
Eq.  (9)). 

There  is  some  distance  below  this  joint,  however, 
where  y  still  remains  equal  to  J/,  while  the  value  of  u 
is  being  determined  from  the  above  Eq.  (39).  Under 
these  circumstances,  /  will  be  found  from  the  following 
after  substituting  the  values  of  y  =  \l,  it  from  Eq.  (39) 
and  A  from  Eq.  (29),  all  in  Eq.  (31). 


(40) 


This  equation  will  be  used  until  a  joint  has  been  reached 
where  the  application  of  q  =  — j —  shows  its  value  to  be 

equal  to  or  greater  than  that  prescribed  for  q.     Here  y 
will  be  determined  by 

2l          l2 


in  which  it  is  seen  to  depend  on  q. 

When   this   point   has   been   reached,   u   will   take   its 

value    from    u=  --  /.  ,    and    y    from    the     equation 


ql2 

~'  w^c^  must   De  substituted  in  Eq.  (31)  to 


determine  /.     This  will  give  after  reduction, 


50 


HIGH    MASONRY    DAM    DESIGN 


All  joints  below  this  point  will  be  found  by  this  last 
equation. 

Summing  up,  we  may  say  that  Eqs.  (34),  (37),  (38),  (40), 
and  (42),  are  the  five  equations  to  be  used  in  determining 
the  length  of  joints  from  the  top  down.  Strictly  speaking 
Eq.  (34)  gives  the  depth  at  which  the  rectangular  portion 
ceases,  while  Eq.  (37)  gives  the  length  of  joints  from  the 
base  of  the  rectangle  down  to  where  y  =  \l\  Eq.  (38)  the 
length  of  joints  from  the  point  where  y  =  \l  to  where  p 
reaches  its  limiting  value;  Eq.  (40)  the  length  of  joints 
from  the  point  where  p  equals  its  limiting  value  to  where 
q  equals  its  limiting  value  and  Eq.  (42)  gives  the  length 
of  all  joints  below. 

Eqs.  (34)  and  (37)  involve  the  value  of  y,  which  is 
obtained  with  respect  to  the  vertical  back,  but  when 
that  face  begins  to  slope  it  is  necessary  to  determine  it 
with  regard  to  the  back  edge  of  the  joint  in  question. 


|Wa 


FIG.  9. 

In  Fig.  9,  mn  represents  the  back  face  of  the  dam  and 
t  is  the  batter  to  be  determined  by  taking  static  moments  of 
A  and  AQ  about  the  back  edge,  m,  of  the  joint. 

The  trapezoid  of  the  figure  is  composed  of  the  triangles 
ht/2  and  (l-l0-t)h/2  and  the  rectangle  hl0. 


HIGH    MASONRY    DAM    DESIGN  51 

t 

By  taking  moments  about  the  edge  w, 


For  Eqs.  (38)  and  (40)  the  value  of  y  must  be,  as  be- 
fore, taken  equal  to  J/,  while  A  has  the  usual  value  of 
A0  +  (l  +  lo)h/2.  Substituting  these  in  Eq.  (43)  and  re- 
ducing: 

(44) 


For  the  joints  to  which  Eq.  (42)  applies  the  value  of  y  is 
to  be  taken  from  Eq.  (41)  as  was  done  before.  In  this 
case: 


By  substituting  this  value  in  the  first  member  of  Eq.  (43) 
and  reducing: 

A0(4l-6y0) 


After  the  value  of  /  is  found  by  the  use  of  Eqs.  (38), 
(40)  or  (42),  /  can  at  once  be  determined  for  the  same 
joint  by  either  Eq.  (44)  or  Eq.  (45). 

In  this  manner  an  entire  theoretical  cross-section  can 
be  determined.  It  will  be  noticed  that  the  location  of  the 
center  of  pressure  in  the  middle  third  of  the  joint  is  the 
governing  condition  in  the  upper  part  of  the  dam,  while 
the  lower  portion  is  fixed  by  the  limiting  pressures  p  and  q. 


52  HIGH    MASONRY    DAM    DESIGN 

The  difficulties  preventing  the  forming  of  a  simple  working 
equation  for  the  entire  cross-section  arise  from  the  fact 
that  the  governing  conditions  are  not  introduced  simul- 
taneously nor  in  the  same  joint. 

By  taking  h  of  the  proper  value,  a  polygonal  cross- 
section  may  be  determined  by  the  preceding  formulae. 
This  cross-section  can  be  then  modified  by  drawing  what 
may  be  called  "mean"  lines,  straight,  broken  or  curved 
along  the  theoretical  faces  so  as  to  adapt  the  latter  to  a 
practical  arrangement  and  treatment  of  the  joints  and 
facing  blocks,  which  may  be  of  cut  stone  or  concrete. 

The  conditions  which  have  governed  the  analysis  are 
essentially  those  of  Rankine,  i.e.,  the  center  of  pressure 
has  in  all  cases  been  kept  within  the  middle  third  of  the 
joint  and  the  greatest  intensity  of  pressure,  either  at  the 
front  face  or  back,  has  not  been  allowed  to  exceed  the 
limit  p  or  q. 


CHAPTER  HI—PART   II 

FORMULA  FOR  DESIGN 
Series  A,  B,  C,  D,  E,  and  F. 

As  noted  earlier  six  separate  series  of  formulae  for 
design  have  been  derived  and  they  will  be  here  set 
forth  in  suitable  form  for  easy  reference  and  use.  As  they 
have  been  developed  by  the  method  just  outlined  it  is 
unnecessary  to  follow  out  the  derivation  of  each  series, 
although  there  exist  some  detailed  differences  in  the  treat- 
ment of  each.  These  details  however,  would  become 
evident  to  anyone  following  the  deductions  throughout. 

Various  conditions  of  "  loading,"  with  approved  as- 
sumptions, such  as  pressure  due  to  expanding  ice  at  the 
water  surface,  upward  water  pressure  on  the  base,  etc., 
referred  to  in  the  table  of  nomenclature  previously  given, 
have  been  introduced  and  are  specifically  stated  for  each 
case. 

It  will  be  recalled  that  Eq.  (31)  is  the  fundamental 
expression ' for  finding  the  length  /,  of  any  joint;  and,  as 
the  several  conditions  are  introduced,  that  the  "  M  "  must 
in  each  case  signify  the  total  overturning  moment  and 
not  merely  the  moment  of  the  static  water  pressure  on 
the  back. 

The  development  of  a  cross-section,  by  any  one  of  the 
following  series,  may  comprise  five  stages,  each  stage 

53 


54 


HIGH    MASONRY    DAM   DESIGN 


representing  the  introduction  of  a  governing  condition. 
Hence,  for  each  stage  there  obtains  a  main  equation  for 
finding  the  length  of  joint  I,  each  main  equation  being 
supplemented  by  secondary  equations  for  y,  u,  and  t\ 
p  and  q. 

It  may  be  necessary  to  employ  more  than  one  of  the 
series  of  equations  in  determining  a  cross-section. 


Flood  Lei 


-*_/ 


FIG.  10. 


For  ready  reference,  the  five  stages  will  be  set  forth 
and  depicted  in  order  as  follows: 

Stage  I. — This  stage,  it  will  be  remembered,  extends 
from  the  top  of  the  dam  to  the  joint  where  the  front 
face  commences  to  batter.  It  is  the  rectangular  section. 
y>\L;  u>%L  (see  Fig.  10).  Ice  pressure  is  purposely 
omitted  in  Fig.  10  to  prevent  confusion  of  letters  in 
small  space.) 


HIGH    MASONRY   DAM   DESIGN 


55 


Stage  II. — This  stage  extends  from  the  lower  limit  of 
Stage  I  to  the  point  where  the  back  face  commences  to 
batter.  u  =  \l;  y^\l  (see  Fig.  u). 

Stage  III. — This  stage  extends  from  the  lower  limit  of 
Stage  II  to  the  point  where  the  intensity  of  pressure  on 
the    toe    has    reached 
the   maximum   allow- 
able intensity.    In  this     Flood  Level 
stage  u  =  |/;  y  =  \l  (see 
Fig.  12). 

Stage  IV.  —  This 
stage  extends  from  the 
lower  limit  of  Stage 
III  to  the  point  where 
the  pressure  intensity 
on  the  heel  has  reached 

the    maximum    allow-      > 

/ 

able   intensity.        For  / 

l_  _     _J-_j- ^ 

this  stage  u>\l\y  =  \l 

(see  Fig.  12). 

Stage  V. — In  this 
stage,  the  limiting  in- 
tensities of  pressure 
at  both  toe  and  heel  having  been  reached,  y>\l\  u>^l. 

This  stage  extends  from  the  lower  limit  of  Stage  IV 
downward.  (See.  Fig.  12). 

The  following  secondary  formulae,  supplementary  to 
the  main  equations  of  all  series,  with  substitutions  as 
noted,  are  arranged  in  order  corresponding  with  the 
preceding. 


FIG.  12. 


56 

Stage  L 

t  =  o 


HIGH    MASONRY    DAM    DESIGN 


L 
ArA 

Stage  II. 


y 


Stage  III. 


t  = 


With  the  condition  of  hydrostatic  up- 
ward pressure  on  the  base  obtaining,sub- 
stitute  the  formulas  in  this  column  in 
place  of  those  correspond  ing,  as  indicated. 


I  2  A  A  rT  \    /  T~ 

P-n—, — cHW2— v 

\   ** 


2JA 


HIGH    MASONRY    DAM    DESIGN 


57 


Stage  IV. 
«  =  §/- 


& 


*  = 


P  = 


Stage  V. 
U--11- 


(limiting 
/     \*      I  I  intensity) 


2  — 


ql2 


u  =  u  — 


«  =  §/- 


t  = 


2  A  f-A  / 
/     \ 


^\  (limiting 
I  )  intensity) 


I 


/       3^\  (limiting 
\        I  )  intensity) 


--)(-? 


If  T  enter  the  following  formulae,  H  above  becomes 
HI.  (See  Figs,  n  and  12.) 

The  first  column  of  formulas  just  given  would  apply, 
with  the  condition  of  upward  pressure  on  the  base  due 
to  hydrostatic  head,  if  a  proper  value  of  u  corresponding,  be 


58  HIGH    MASONRY    DAM    DESIGN 

taken,  that  is  if  the  excursion  of  the  force  AAf,  resulting 
from  the  effect  of  all  other  forces  on  AAf  be  considered, 
rather  than  the  effect  of  all  the  other  forces  on  the  re- 
sultant vertical  force. 

It  will  be  found  expeditious  to  design  a  section,  where 
ice  pressure  at  the  level  of  full  reservoir  is  to  be  considered 
in  connection  with  the  water  surface  at  some  higher  flood 
level,  first  by  series  of  formulas  containing  T  (cf .  Series  B  and 
D)  and  then  to  investigate  successive  bases,  or  joints,  thus 
obtained  (beginning,  for  a  high  masonry  dam,  usually 
at  a  base,  or  joint,  about  100  feet  from  the  top  of  the  dam) 
with  series  of  formulae  lacking  T,  or  the  ice  pressure  con- 
dition (cf.  Series  A  and  C).  A  base  will  ultimately  be 
obtained  by  these  supplementary  "  Flood  level  "  calcula- 
tions greater  than  the  base  at  its  same  elevation  as  pre- 
viously determined  by  the  "  Ice  Pressure  "  design. 

Continuing  with  the  design  by  means  of  the  "  Flood  level  " 
formulae  to  the  bottom  of  maximum  height  required  will 
determine  the  minimum  cross-section  area  to  meet  the 
conditions  both  of  "  Flood  "and  of  "  Ice."  It  should  be 
remarked  in  this  connection  that  when  a  reservoir  level 
is  rising  due  to  flood  conditions  prevailing,  it  is  evident 
that  ice  formation  cannot  develop,  or,  in  other  words,  the 
two  conditions  cannot  be  coexistent,  hence  the  difference 
in  designation  of  hydrostatic  heads  corresponding.  (See 
Figs,  ii  and  12.) 


HIGH    MASONRY    DAM    DESIGN  59 


SERIES  A. 

Conditions:     Overturning    moment    due    to    horizontal 
static  water  pressure  on  back  of  dam  only. 

Stage  I. 


Siape  II. 

12  + 


Stage  III. 


Stage  IV. 


P 

Stage  V. 


SERIES  B. 

Conditions:    Overturning  moment  due  to: 

(a)  Horizontal  static  water  pressure  on  back  and 

(b)  Ice  pressure  applied  at  distance  (a\)  from  top, 


Stage  I. 


60  HIGH    MASONRY    DAM    DESIGN 

Stage  II. 


Stage  HI. 


Stage  IV. 

P  =  (Hi3  +  6rHi)£. 

Stage  V. 


SERIES  C. 

Conditions:    Overturning  moment  due  to: 

(a)  Horizontal  static  water  pressure  on  back. 

(b)  Upward  water  pressure  on  base.     Pressure  intensity 
decreasing  uniformly  from  cH-y  at  heel  to  zero  intensity 
at  toe. 

Stage  I. 
H  = 

Stage  II. 


Stage  III. 
cH 


HIGH    MASONRY    DAM    DESIGN  61 


Stage  IV. 

cH 


p   \  h 


J  which  reduces  to 


P' 

Stage  V. 

cH 


~  +  ^o5  which  reduces  to 


SERIES  D. 

Conditions:    Overturning  moment  due  to: 

(a)  Horizontal  static  water  pressure  on  back  (head  =  H\). 

(b)  Ice  pressure  applied  at  distance  (a\)  from  top. 

(c)  Upward  water  pressure  on  base.     Pressure  decreasing 
uniformly  from  cH\f  at  heel  to  zero  intensity  at  toe. 

Stage  I. 
Hl= 

Stage  II. 


62  HIGH    MASONRY    DAM    DESIGN 

Stage  III. 


Stage  IV. 


1){2A0       \ 
-  I  -7—  +  /o  )  , 


which  reduces  to 


Stage  7. 


2A0  ,  ,\ 
-T—  +  /O  I, 


which  reduces  to 


In  the  preceding  series  of  equations  it  will  be  observed 
that  the  final  expressions  for  /  in  stages  IV  and  V  are 
very  similar,  and  that  the  quantity  c  in  equations  (a)  of 
these  stages  disappears  in  equations  (b).  Equations  (b) 
of  course,  are  to  be  used  for  purposes  of  calculation  of 
cross-sections. 


HIGH    MASONRY    DAM    DESIGN 


63 


SERIES  E. 

Conditions:    (See  Fig.  13.)    Ice   pressure  neglected  in 
Fig.  13.     Overturning  moment  due  to: 

(a)  Horizontal  static  water  pressure  on  back  (head  =  /*i). 

(b)  Ice  pressure  applied  at  distance  (a\)  from  top. 

(c)  Upward  water  pressure  on  base;    pressure  intensity 
decreasing  uniformly  from  c(hi+h2)r  at  heel  to  zero  in- 
tensity at  toe. 

(d)  Mud  (liquid)  pressure  on  back  (head  h2) ,  commencing 
at  distance  h2  above  joint  in  question.     Weight  of  mud  =  f . 
As  before,  if  T  be  equated  to  zero,  a\  becomes  equal  to 
a,  in  the  formulas. 


FIG.  13. 
Stage  I.     (hi,  of  known  value,  h2  to  be  determined.) 

feY , 


Stage  II. 


64  HIGH    MASONRY    DAM    DESIGN 

For  trapezoidal  section  at  top,  make  AQ  =  o  and  yo 
and  IQ=L  in  Stage  II.     This  applies  generally. 

Stage  III. 


Stage  IV. 


Stage  V. 


From  a  study  of  the  formulae  thus  far  developed  it 
will  be  observed  that  by  reducing  certain  conditions  to 
zero,  with  their  corresponding  quantities,  the  main  equa- 
tions of  a  given  series  reduce  to  those  of  a  simpler  series. 

For  instance- 

In  Series  B  make  T  (for  ice  pressure  condition  of  load- 
ing) equal  to  zero  and  H\=H  and  a\=a  and  the  main 
equations  of  that  series  reduce  to  Series  A  equations. 

In  Series  C,  by  making  c  (for  upward  water  pressure 
condition)  equal  to  zero  in  main  equations  of  Stages  I,  II, 
and  III  and  also  in  equations  (a)  of  Stages  IV  and  V,  the 
equations  of  Series  C  reduce  to  those  of  Series  A. 

Likewise,  by  making  the  proper  eliminations  and  sub- 
stitutions, Series  E  will  reduce  to  Series  D,  C,  B  or  A. 


HIGH    MASONRY    DAM    DESIGN 


65 


SERIES  F. 

This  series  consists  of  general  formulae  for  a  number 
of  imposed  conditions  of  loading.  For  any  given  case, 
the  terms  or  factors  expressing  those  conditions  not  ap- 
pertaining must  be  eliminated  by  equating  them  to  zero. 
(See  Fig.  14.) 


FIG.   14. 


Conditions  for  General  Formula. 

Overturning  moment  due  to: 

(a)  Horizontal  static  water  pressure  on  back  (head  =  /*i). 

(b)  Upward  water  pressure  on  base;    pressure  intensity 
decreasing  uniformly  from  cHf  or  c(/h-f^2)r>  a^  nee^  to 
zero  intensity  at  toe. 

(c)  Mud  (liquid)  pressure  on  back  (head  h2)  as  before. 

(d)  Dynamic  pressure  of  water,  Df. 

(e)  Water  flowing  over   top   of  dam,   weight  of  water, 
of  depth  6,  on  top  of  dam  being  neglected. 


66  HIGH    MASONRY   DAM    DESIGN 

For  condition  of  water  not  overtopping  dam,  b  =  o  and 
£>=o.* 

For  condition  of  no  dynamic  pressure,  D  =  o. 

For  condition  of  no  upward  water  pressure,  c  =  o. 

For  condition  of  no  mud  (i.e.,  mud  being  replaced  by 
water)  make  h2  =  o,  h\=H. 

Stage  I. 

Rectangular  cross-section  at  top  or  rectangular  dam, 
l=k=L. 

This  may  fall  under  either  of  two  cases,  viz.  — 

Case  (i) 

Condition:  hi=H\  h2  =  o. 


Case  (2) 

Condition:  hi  of  known  value;  h2  to  be  determined. 


As  in  the  preceding  series,  the  value  of  H  or  h2,  of 
Stage  I  may  be  determined  by  several  successive  trial 
substitutions. 

Stage  II. 

(a)  Trapezoidal  cross-section  at  top  of  dam  or  trape- 

*  For  water  surface  below  top  of  dam,  Series  E,  containing  the  distance  'a" 
must  be  used. 


HIGH    MASONRY    DAM    DESIGN  67 

> 

zoidal  dam  (spillway)  front  face  battered.     (A0  =  o,  10=L 
and  >>o  =  o.)     Note:    For  a  triangular  dam  /0  =  o,  also. 


(b)  Trapezoidal  section  continued  (front  face  battered). 


Stage  III.  —  Both  faces  battered. 


O/ 


Stage  IV.  —  Limiting  intensity  of  pressure,  p,  introduced. 


Stage  V.  —  Limiting  intensities,  p  and  q. 


68 


HIGH    MASONRY    DAM    DESIGN 


The  increased  number  of  overturning  loads  then,  tend 
to  render  the  right-hand  members  of  the  various  equations 
more  involved;  though  after  a  little  practice  one  may 
easily  carry  through  a  design  with  surprising  rapidity. 
The  slide  rule  may  be  used  to  great  advantage  and  it  is 
suggested  that  the  results  be  tabulated  as  determined, 
in  some  such  form  as  the  following: 


TABLE  OF  RESULTS. 


c=   ,  T=    ,  etc. 


Joint 
No. 

H 

H« 

h 

H  +  a 

A0 

A 

h 

/o2 

/ 

yo 

y 

u 

t 

p 

Q 

Aoyo 

Etc. 

I 

2 

3 

4 

Etc. 

CHAPTER   IV 
FORMULA    FOR  INVESTIGATION 

THE  effect  upon  the  calculation  of  a  cross-section,  of 
backfill  on  the  down-stream  face,  could,  of  course,  be  cared 
for  by  introducing  that  condition  into  the  preceding  series 
of  equations;  but  as  this  effect  as  computed,  would  be, 
in  any  case,  largely  dependent  upon  assumptions  which 
may  vary  widely  and  as  the  placing  of  backfill  is  generally 
a  later  consideration  with  respect  to  construction,  the 
propriety  of  such  introduction  at  that  stage  of  design  is 
questionable. 

In  the  following  formulae  for  investigation  therefore, 
the  general  conditions  of  an  earth  thrust  acting  at  the 
down-stream  face  and  of  a  vertical  component  of  thrust  of 
material  on  the  up-stream,  inclined  face  of  the  dam,  are 
introduced.  Moments  of  forces  are  taken  about  a  point 
in  the  joint  distant  y  from  the  up-stream  edge  of  the  joint 
in  derivations  for  u. 

By  any  of  these  formulae  the  position  of  the  line  of 
resistance  for  any  given  cross-section  and  respective  con- 
ditions may  be  determined  with  regard  to  any  horizontal 
joint  and  its  down -stream  edge;  the  value  of  u  being  the 
quantity  to  be  sought. 

Any  condition  may  be  disregarded    by    equating  its 

term  to  zero. 

69 


70  HIGH    MASONRY   DAM    DESIGN 

The  first  expression  below  contains  all  of  the  conditions 
heretofore  considered  with  the  additional  ones  just  stated; 
and  from  it  follow  the  succeeding  expressions  for  u.  It 
should  be  remembered  that  the  term  T  cannot  be  coex- 
istent in  any  expression  for  stability  with  b  and  therefore 
with  D.  Nevertheless  all  of  these  terms  are  written  with 
the  understanding  that  the  proper  eliminations  be  always 
made.  Three  general  group  equations  will  be  written. 

Formula  for  Investigation. 

First,  Conditions  of  retained  mud,  water,  overtopping, 
etc.  (see  Fig.  14). 


f  2h2-b)-6Wv(y-s) 

—  y)  sin  d  — : — 

3        sin  a     J  ) 


U~      y 

Whence,  for  conditions  of  retained  mud,  water,  etc.,  but 
no  overtopping,  by  making  b  =  o  and  D  =  o,  there  follows 
(see  Fig.  13): 


From  this  last  expression  for  u,  for  conditions  of  re- 
tained water,  etc.,  but  neither  mud  nor  overtopping,  by 
making  hi=Hi\  h2  =  o,  there  is  obtained: 


HIGH    MASONRY    DAM    DESIGN  71 

i 


+r(j-y)sin,_ 

„_/     y 


As  in  the  equations  for  design,  when  7  =  o,  HI  =H.  (See 
Figs.  1  1  and  12.)  If  H'  is  of  such  depth  that  the  down-stream 
batter  of  the  cross-section  varies  considerably,  an  approxi- 
mate solution  is  possible  by  assuming  some  average  batter 
for  the  lower  portion.  The  expression  for  earth  thrust  is 
general,  as  is  evidenced.  After  u  is  determined  for  each  joint, 
the  intensities  of  maxima  pressures  can  be  determined 
for  the  given  cross-section,  the  general  expression  for  p, 
corresponding  to  above  expressions  for  u,  being: 


In  connection  with  the  computation  for  the  value  of  y  in 
an  investigation,  as  indicated  above,  it  is  necessary  to  obtain 
the  position  of  the  centroid  of 
a  trapezoid  with  respect  to  the  ~~l~° 

back,    or   up-stream   edge,    of  jn 

the  joint  in  question.     The  fol-         j\  centroid-f- 
lowing  expression  for  x,  in  con- 
nection with  Fig.  15,  may  prove 

convenient:  f~"  ~~l'~          ~~*\ 

FIG.  15. 


j_  -x  _____  >, 


It  is   desirable   to   consider   tension   as   active  in   the 
joint,  and  if  p't  is  the  intensity,  in  tons  per  square 


72  HIGH   MASONRY    DAM    DESIGN 

at  the  down-stream  end  of  the  joint,  and  p"t  is  the  inten- 
sity at  the  up-stream  edge  of  the  joint,  p,  above,  which,  as 
written,  is  in  pounds  per  square  foot,  will  take  the  form : 


and 


In  the  case  where  there  is  liquid  mud  only  on  the  back, 

Af  r' 
Wv  becomes  equal  to  — '—,  where  A'  is  the  area  of  the 

r 

superimposed  mud. 

In  the  use  of  the  foregoing  formulae,  it  may  be  desirable 
to  take  account  of  some  such  variation  of  the  extent  of  the 
uplift  intensity  on  the  base  or  joints,  as  was  indicated 
in  Chapter  I,  p.  14,  in  the  distinction  to  be  drawn  be- 
tween uplift  conditions  in  the  foundations  and  higher 
up  in  the  dam. 

For  the  foundation,  a  trapezoidal  distribution  was  sug- 
gested and  for  the  body  of  the  masonry,  drained  by  wells, 
a  triangular  distribution  of  intensities,  but  of  shortened 
extent  down-stream,  was  proposed. 

In  the  former  case,  the  total  pressure  would  be  in- 
creased, but  its  lever  arm  would  tend  to  be  diminished. 
In  the  latter  case,  the  pressure  would  be  diminished,  but 
the  lever  arm  increased.  The  effect  on  a  cross-section 
design,  or  on  a  line  of  pressure  for  a  given  cross-section, 
would  have  to  be  worked  out  for  any  particular  case.  For 
such  a  structure  as  the  Kensico  Dam,  the  foregoing  changes 
from  the  ordinary  triangular  assumption,  while  modifying 
the  numerical  results  as  to  ''factors"  against  overturning 


HIGH    MASONRY    DAM    DESIGN  73 

f/  / 

and  as  to  resulting  pressure  Intensities  on  the  various  joints 
and  the  foundation,  did  not  modify  the  final  cross-section. 

An  approximation,  however,  may  be  satisfactorily  ob- 
tained by  varying  the  value  of  c  as  one  works  down  the 
cross-section  from  the  top.  The  amount  of  the  requisite 
variation  may  be  ascertained  by  comparing  the  overturning 
moment  of  the  uplift,  acting  as  usually  assumed,  with  the 
value  of  the  overturning  moment  as  desired  to  be  assumed 
in  the  given  case.  This  may  be  done  for  three  different 
points  down  the  dam,  and  the  ratio  found  by  this  com- 
parison. A  curve  may  then  be  plotted  in  terms  of  these 
ratios,  and  the  distances  from  the  top,  all  figured  from  a 
cross-section  assumed  as  nearly  like  the  dam  under  con- 
sideration as  can  be  anticipated.  A  curve  through  the 
three  points  will  yield  the  relative  values  of  the  variable 
c's  to  be  used  in  the  formulae,  each  c  for  its  own  elevation. 
Or  else  formulae  such  as  these  here  given  may  be  derived 
for  the  given  assumption  and  used  directly. 

In  the  studies  for  design,  referred  to  before,  the  analytic 
work  should  be  checked  throughout  by  the  graphic  method 
wherever  possible.  This  should  always  be  done  both  in 
designing  and  investigating  cross-sections. 

It  should  be  stated  here  that,  after  a  cross-section  has 
been  fixed  upon  for  a  given  dam  and  the  faces  drawn  to 
chosen  batters  and  curves,  the  entire  cross-section  should 
be  investigated  as  just  outlined  so  as  to  give  the  actual 
values  for  this  final  cross-section. 

Again,  in  comparing  different  cross-sections,  especially 
of  different  dams,  by  superimposing,  their  water  lines 
should  be  made  to  coincide  and  not  their  tops  for  a  fair 
comparison. 


CHAPTER  V 
THE   DESIGN  OF  A  HIGH   MASONRY  DAM 

To  illustrate  the  method  of  applying  the  preceding 
formulae  to  the  determination  of  the  theoretical  cross- 
section  of  a  high  masonry  dam,  an  actual  problem  will 
be  presented.  For  this  purpose  the  Olive  Bridge  Dam 
has  been  selected,  not  only  because  it  is  representative  of 
the  type  for  which  the  formulae  were  developed,  but  be- 
cause the  structure  has  been  recently  put  into  service, 
and  is  sufficiently  well  known  to  be  of  more  than  passing 
interest. 

It  may  not  be  inappropriate,  before  proceeding  to  the 
computations,  to  refer  to  certain  of  the  structure's  more 
important  features,  especially  as  some  were  entirely  new, 
and  to  give  a  brief  description  of  it. 

The  Olive  Bridge  Dam  is  the  principal  structure  of  a 
number  of  dams  and  dikes  which  serve  to  impound  the 
waters  of  the  Ashokan  Reservoir.  The  latter  is  located 
about  14  miles  west  of  the  Hudson  River  at  Kingston, 
N.  Y.,  has  an  available  storage  capacity  of  128  billion 
gallons,  derived  from  the  Esopus  watershed,  with  an  area 
of  255  square  miles,  and  delivers  the  stored  water  to  the 
Catskill  Aqueduct,  whose  capacity  is  500,000,000  gallons 
per  day,  to  be  conducted  to  the  City  of  New  York,  about 
100  miles  away,  and  on  the  opposite  side  of  the  Hudson 

River. 

74 


HIGH    MASONRY    DAM    DESIGN  75 

i 

The  reservoir  is^formed^by  the  Olive  Bridge  Dam, 
placed  across  Esopus  Creek,  and  by  the  West,  Middle, 
East  and  Hurley  dikes,  located  in  the  smaller  gaps  between 
the  hills  which  create  the  natural  basin. 

The  dividing  dike  with  its  weir  separates  the  reservoir 
into  two  portions,  while  the  waste  weir,  nearly  1000  feet 
long,  at  the  eastern  end  of  the  large  dikes,  provides  the 
means  of  discharging  surplus  floods  safely.  All  of  the 
weirs  are  masonry  structures. 

The  length  of  the  reservoir  is  12  miles,  with  a  max- 
imum width  of  3  miles,  while  the  total  length  of  the  dam 
and  the  dikes  is  5^  miles. 

Work  was  started  in  the  latter  part  of  1907,  and  on 
Sept.  9,  1913,  the  storage  of  water  in  the  west  basin  began. 
By  Oct.  2  in  the  same  year,  the  first  Esopus  water  could 
have  been  delivered  by  gravity  into  the  Catskill  Aqueduct, 
the  water  surface  of  the  west  basin  having  reached  ele- 
vation 495,  or  a  depth  of  about  95  feet  behind  the  dam, 
the  equivalent  of  2100  million  gallons  impounded,  but 
unavailable. 

Three  types  of  dam  were  considered  in  the  studies. 

(1)  An  earth  dam,   to  be  constructed  by  sluicing  or 
some  other  method. 

(2)  A  composite  dam,  or  one  consisting  of  a  masonry 
core,    covered    by    an    earth    embankment,    the    masonry 
portion  being  of  small  section  across  Esopus  gorge  and 
rising  to  within  50  feet  of  the  water  surface  for  full  reser- 
voir, the  earth  embankment  making  up  the  remainder. 

(3)  A   masonry   dam  of   the   gravity   type,   extending 
across  the  gorge  and  flanked  by  earth  wings  at  each  side 
of  the  valley. 


76  HIGH    MASONRY    DAM    DESIGN 

The  investigations  were  so  far  advanced  at  the  end  of 
1906  that  a  decision  in  favor  of  the  third  type  was 
reached  early  in  1907,  after  268  study  drawings  had  been 
made. 

The  masonry  dam  is  founded  on  solid  ledge  rock.  A 
cut -off  trench  extends  about  40  feet  below  the  stream 
bed,  and  grout  holes  go  20  feet  deeper.  The  main  struc- 
ture is  of  cyclopean  masonry,  with  concrete  face  blocks, 
while  the  wings,  built  of  acceptable  earth  found  near  the 
site,  contain  concrete  corewalls  extending  to  solid  ledge 
or  into  very  compact  impervious  earth. 

The  masonry  portion  of  the  main  dam,  which  rises  to 
an  elevation  of  210  feet  above  the  stream  bed,  is  1000  feet 
long,  but  the  total  length  of  the  structure  including  the 
earth  wings  is  4650  feet. 

To  prevent  temperature  cracks  in  the  masonry  section, 
it  was  decided  to  divide  the  dam  transversely  by  means 
of  vertical  "  expansion  joints,"  the  distance  between  which 
would  be  well  within  those  distances  at  which  cracks  had 
heretofore  been  observed  in  other  structures.  These  in- 
tervals varied  from  84  to  91  feet. 

The  expansion  joints  (more  properly  termed  contraction 
joints)  are  formed  by  building  vertical  faces  of  concrete 
blocks,  shaped  as  a  tongue-and-grooved  joint,  normal  to  the 
axis  of  the  dam,  and  thus  preventing  a  continuous  opening 
through  the  structure,  should  the  adjacent  sections  contract. 
Vertical  inspection  wells  at  each  expansion  joint  afford  the 
opportunity  of  studying  the  conditions  at  these  sections. 

There  are  also  two  longitudinal  inspection  galleries 
built  within  the  dam,  one  near  the  top  and  entered  by 
manholes  from  the  surface,  and  one  near  the  lower  portion 


HIGH    MASONRY    DAM    DESIGN  77 


of  the  dam,  both  of  which  ^are  connected  with  the  vertical 
inspection  wells. 

The  inspection  wells  afford  the  opportunity,  should  it 
be  deemed  desirable,  of  placing  a  copper  strip  across 
each  expansion  joint,  to  reduce  the  quantity  of  water  that 
may  be  expected  to  pass  through  them. 

The  lower  longitudinal  gallery  opens  near  the  center 
of  the  dam  into  a  transverse  gallery,  leading  to  a  measur- 
ing weir  chamber  and  drain,  at  the  downstream  side,  where 
the  entire  leakage  may  be  gaged  and  discharged. 

Between  the  vertical  inspection  wells,  drainage  wells 
1  6  inches  in  diameter,  and  about  12  feet  apart,  slightly 
inclined  downstream  from  the  top,  are  provided,  between 
the  upper  and  lower  longitudinal  galleries,  to  intercept 
seepage  into  the  masonry  and  to  prevent  any  water  from 
reaching,  and  consequently  disfiguring,  the  downstream 
face  of  the  structure. 

These  wells  were  constructed  by  laying  up  large,  hollow, 
porous  concrete  blocks. 

Small  quantities  of  water  which  may  enter  the  body 
of  the  dam,  either  through  the  expansion  joints,  into  the 
inspection  wells,  or  through  the  capillary  spaces  in  the 
masonry,  will  be  conducted  by  means  of  the  wells,  galleries 
and  drains  to  the  gorge  below  the  dam. 

It  is  proposed  later  to  fill  the  vertical  inspection  wells 
with  material  that  will  effectively  stop  all  flow. 

The  following  calculations  indicate  in  detail  the  method 
pursued  in  the  determination  of  the  theoretical  cross- 
section  of  the  Olive  Bridge  Dam. 

The  conditions  that  governed  in  the  design  were  as 
follows  : 


78  HIGH    MASONRY    DAM    DESIGN 

1.  Elevation  of  top  of  dam  above  datum 610   ft. 

2.  Elevation  of  free  water  surface,  reservoir  full 590    ft. 

3.  Elevation  of  free  water    surface,   reservoir  in  max. 

flood 596    ft. 

4.  Elevation  of  free  water  surface,   reservoir  in  max. 

flood  and  wind  blowing  at  max.  velocity  of  40 

miles  per  hour,  with  waves  piled  up 600    ft. 

5.  Elevation  of  bottom  of  max.  section ;  rock  excavation 

at  this  point  approximately  10  feet  deep 390    ft. 

6.  Ice  pressure,  assumed  the  same  as  that  used  in  the 

design  of  Wachusett  Dam,  applied  at  elevation  590.  23.5  tons  per  sq.ft. 

7.  Upward  pressure  due  to  hydrostatic  head  assumed 

to  be  of  uniformly  varying  intensity,  and  varying 
from  a  max.  at  the  heel  of  the  joint  to  zero  at  the 
toe,  distributed  over  f  the  area  of  the  joint. 

8.  At  the  maximum  section  the  earth  is  refilled  to  the 

top  of  the  gorge  equivalent  to  elevation  450  or  500. 

9.  Other  conditions  so  liberal  that  section  will  be  reason- 

ably safe  against  earthquake  and  dynamite. 

10.  Maximum  allowable  unit  pressure 20  tons  per  sq.ft. 

11.  Top  width 23  ft. 

12.  Resultant  line  of  pressure,  reservoir  full  and  empty, 

shall  lie  within  the  middle  third  at  each  joint. 

13.  4,  or  ratio  of  unit  weight  of  masonry  to  unit  weight 

of  water £ 

14.  fy  weight  of  a  cubic  foot  of  water 62 . 5  Ibs. 

15.  c,  ratio  of  upward  thrust  intensity  due  to  hydrostatic 

head,  assumed  to  act  at  heel  of  joint f 

Flood  conditions,  Series  C,  and  ice  conditions,  Series  D, 
will  be  imposed,  and  the  design  prosecuted  with  respect 
to  each  simultaneously. 

From  the  above  it  follows  that 

a  =  10  feet  (flood  conditions), 
ai  =20  feet  (ice  conditions), 
67  =  4512, 

C  _2 

A~7' 

JOINT  No.  i,  FLOOD  CONDITIONS 

Generally  speaking  it  may  be  assumed  that  the  top  of 
a  masonry  dam  will  be  about  TV  of  the  height  above  the 


HIGH    MASONRY    DAM    DESIGN  79 

i 

water,  but  in  the  case^  under  consideration,  a  superelevation 
of  only  20  feet  was  employed  for  full  reservoir,  and  10  feet 
for  flood  conditions.  While  the  choice  in  this  respect  is 
purely  arbitrary,  the  above  ratio  is  the  one  usually  pre- 
scribed if  there  are  no  other  governing  conditions. 

Since  the  length  of  a  joint  depends  upon  its  depth 
below  the  water  surface,  it  is  evident  that  at  the  surface 
this  dimension  should  be  zero.  For  various  reasons,  how- 
ever, such  as  the  desirability  of  a  foot- walk  or  a  driveway 
on  the  crest,  a  top  width  is  chosen  which  will  satisfy  the 
demands. 

As  23  feet  had  been  decided  upon  as  the  top  width, 
for  a  considerable  distance  below  the  water  surface  the 
rectangular  section  will  more  than  satisfy  the  only  con- 
dition for  stability  that  applies  in  this  portion  of  the  dam, 
namely,  that  the  resultant  of  all  the  external  forces  for  res- 
ervoir full  and  reservoir  empty  shall  be  within  the  middle 
third  of  the  cross-section.  It  is  evident  that  for  reservoir 
empty,  the  resultant  passes  through  the  center  of  the 
joint.  It  becomes  necessary,  however,  for  reservoir  full, 
to  determine  the  depth  H  at  which  this  resultant  first 
emerges  from  the  middle  third  of  the  rectangular  section. 

For  flood  conditions  we  will  use  the  equation  under 
Stage  I,  Series  C,  which  is, 


Here  a  =  10  feet,  A  =  J,  L  =  23  feet ;  and  c  =  f . 

This  equation  may  be  solved  by  successive  substi- 
tions  for  H,  until  such  a  value  of  H  is  found  that  equality 
results.  In  the  present  instance  it  is  found  that  #=35.1 
feet  satisfies  the  equation,  and  hence  the  rectangular  cross- 


80  HIGH    MASONRY    DAM    DESIGN 

section  of  the  dam  may  be  carried  down  to  a  depth  of 
45.1  feet  below  the  top,  since  a,  the  superelevation  under 
flood  conditions,  is  equal  to  10  feet. 

JOINT  No.  ia,  ICE  CONDITIONS 

In  a  similar  manner  for  ice  conditions,  we  must  use 
the  following  formula,  under  Stage  I,  Series  D,  to  deter- 
mine the  depth,  Hi,  at  which  the  resultant  first  emerges 
from  the  middle  third  of  the  rectangular  section. 


Here  a\  =  20  feet,  A  =  J,  L  =  23  feet,  c  =  f  and  6T  =  45 1 2. 

By  successive  substitutions  for  Hi,  it  is  found  that  a 
value  of  6.7  feet  will  satisfy  the  equation  and  consequently 
the  rectangular  section  in  this  case  can  be  carried  down 
only  6.7  feet  below  full  reservoir,  or,  since  the  superelevation 
a\  is  20  feet,  to  a  depth  of  only  26.7  feet  below  the  top, 
before  it  will  be  necessary  to  modify  the  section. 

A  comparison  of  the  two  values  established,  HI  -\-a\  = 
26.7  feet,  and  H-\-a  =  4$.i  feet,  together  with  an  exam- 
ination of  the  profile,  shows  the  very  marked  effect  the 
assumption  of  ice  pressure  has  upon  increasing  the  cross- 
section  in  the  upper  levels  of  the  dam. 

The  solution  for  either  H  or  HI  may  be  expedited  by 
the  use  of  the  graphic  method.  Thus,  assume  at  least 
three  values  for  H,  say  30  feet,  40  feet,  and  50  feet  in 
the  present  case,  substitute  successively  in  the  right-hand 
member  of  the  above  equation  and  solve.  Plot  these 
resulting  values  as  abscissae  and  the  assumed  values  for 
H  corresponding  as  ordinates.  A  smooth  curve  drawn 
through  the  points  thus  obtained  will  give  a  point  where 


HIGH    MASONRY   DAM   DESIGN  81 

ordinate  and  abscissae  are  e%ial,  and  this  will  locate  the 
desired  value. 

At  no  point  in  this  portion  of  the  dam  does  the  length 
of  a  horizontal  joint  change,  but  below  a  depth  of  26.7 
feet  from  the  top,  this  dimension  will  have  to  be  increased 
in  order  to  comply  with  the  requirement  for  stability,  which 
prescribes  that  the  resultant  of  all  external  forces  shall 
lie  within  the  middle  third.  This  is  accomplished  by 
giving  a  batter  to  the  downstream  face  of  the  dam,  while 
the  upstream  face  remains  vertical. 

This  stage  of  the  design  extends  from  the  lower  limits 
of  the  rectangular  section  to  that  elevation  where  it  first 
becomes  necessary  to  batter  the  back,  and  -the  formulae 
to  be  used  are  those  found  under  Stage  II,  Series  C,  for 
flood,  and  Series  D  for  ice  conditions. 

JOINT  No.  2,  FLOOD  CONDITIONS 

The  investigation  for  the  purpose  of  determining  the 
length  /  of  joint  No.  2  involves  the  use  of  an  equation  in 

which  u  shall  have  a  value  of  -,  since  the  resultant  of  the 

3 
external  forces  for  the  reservoir  full  reached  the  limit  of 

the  middle  third  at  the  downstream  side  at  joint  No.  i, 
and  since,  also,  it  may  not  pass  outside  that  limit.  This 
is  expressed  for  flood  conditions  by  the  following  equation 
from  Stage  II,  Series  C  : 


The  value  of  4.9  feet  will  be  given  to  h,  to  bring  the 
depth  of  joint  No.  2  to  elevation  560,  or  40  feet  below  the 


82  HIGH    MASONRY   DAM    DESIGN 

flood  level,  and  50  feet  below  the  top  of  the  dam,  merely 
because  this  is  an  easier  figure  to  work  with  and  because 
the  depth  of  each  successive  joint  is  taken  as  10  feet,  or 
some  multiple  of  10  feet,  below  the  next  above. 

[n  the  above  expression  the  factors  take  the  following 
values  : 

•^0  =  1035  square  feet;    #  =  40  feet;    ^=4.9  feet; 

\    and       A=$. 


Substituting  these  values  in  the  above,  completing  the 
square,  and  solving  for  /,  we  obtain  : 

/  =  25  feet. 

As  it  is  necessary  to  use  the  factor  A  at  each  successive 
joint,  it  should  be  determined  at  this  point 


A  = 


All  the  quantities  in  the  above  equation  are  known, 
and  A  is  found  to  equal  1152.5  square  feet. 

It  will  be  noted  that  the  equation  employed  to  deter- 
mine the  value  of  /  involves  the  factor  yo,  which  represents 
the  distance  from  the  up-stream  face  to  the  point  of  appli- 
cation of  the  resultant  pressure  for  reservoir  empty,  on 
the  joint  next  above  the  one  being  investigated.  It  there- 
fore becomes  necessary  after  each  application  of  the  formula 
to  use  the  supplementary  equation  from  which  the  value 
of  y  is  found,  not  only  for  the  above  purpose  of  locating 
the  point  of  application  of  the  resultant,  reservoir  empty, 
but  because  this  y  becomes  in  turn  yQ  for  the  joint  next 
below. 


HIGH   MASONRY;  DAM    DBSIGN  83 

Furthermore,  as  ir  has  already  been  given  the  value 

of  -,  the  only  condition  for  stability  in  this  Section  of  the 
o 

dam  involving  Stage  II  is  that  y  shall  be  equal  to  or  greater 

than-. 
3 


As  all  of  these  factors  are  already  known,  it  merely 
requires  that  they  be  substituted  in  the  above,  which 
results  in 

y*n.6  feet, 

This  indicates  that  the  pressure  line  P',  for  reservoir 
iMiipty,  lies  well  within  the  middle  third  of  Joint  No.  2, 

As  the  limiting  depth  of  the  rectangular  section  under 
ice  conditions  was  found  to  be  26.7  feet  below  the  top  of 
the  dam,  while  for  flood  conditions  it  was  located  45.1 
feet  below  the  top,  it  will  be  necessary  to  investigate  the 
joint  at  the  latter  point  and  called  joint  No.  i,  under  the 
ice  conditions,  to  see  what  effect  the  ice  will  have  upon  its 
length.  After  the  J  has  been  determined  for  this  joint, 
the  length  of  the  joint  under  ice  conditions  at  a  depth 
of  50  feet  below  the  top,  called  joint  No.  2,  will  be  found, 
so  that  both  joint  No,  i  and  joint  No.  2  will  have  values 
of  /  undtM-  flood  i-oMilitions  .nul  under  ice  conditions,  and 
thereafter  the  procedure  will  be  to  investigate  the  length 
of  each  successive  joint,  first  ,  for  flood  and  then  for  ice. 

The  following  formula  from  Stage  II,  Series  D,  will 
be  employed: 


84  HIGH    MASONRY    DAM    DESIGN 

JOINT  No.  i,  ICE  CONDITIONS 


In  this  expression  the  following  factors  are  known: 
c  =  f,  A=f,  7/1=25.  i  feet,  /*  =  i8.4  feet,  AQ=6i$  square 
feet,  57=4512,  y0  =  ii.$  feet,  /o  =  23  feet;  whence,  by 
substitution,  completing  the  square,  and  solving  for  /, 
we  obtain  the  value 

£=33.0  feet. 
To  determine  A,  we  have 

A  =A0+~  —  h  =  1130  square  feet. 

The  preceding  determination  of  the  value  of  /,  shows 
that  at  a  depth  of  45.1  feet  below  the  top  of  the  dam 
the  length  of  the  same  joint  should  be  23  feet  under  flood 
conditions,  and  33  feet  when  ice  is  assumed  to  act. 

We  must  now  determine  y,  since  its  value  must  be  used 
in  the  joint  next  below  where  it  takes  the  symbol  yQ,  and 
the  following  formula  is  employed,  in  which  all  the  quan- 
tities are  known  : 


12.7  feet. 


It  should  be  noted  here  that  the  denominator  in  the 
above  is  equal  to  A. 
Similarly  for 


HIGH    MASONRY --DAM   DESIGN  85 

JOINT  No.  2,  ICE  CONDITIONS 

Using  the  same  formulae  for  I  as  in  joint  No.  i  with 
new  values  to  some  of  the  factors,  thus : 

H!  =  30  feet,  h  =  4. 9  feet,  /0=33  feet,  ^0  =  1130  square 
feet,  ^0  =  12.7  feet. 

We  find  that 

^=35-5  feet. 

Computing  the  value  of  A 

A  =A0-\-^-h  =  1298  square  feet, 

and  solving  for  y  with  the  factors  from  above  and  using 
A  =1298  square  feet  for  the  denominator,  we  obtain 

y  =  i3-3  feet. 

JOINT  No.  3,  FLOOD  CONDITIONS 

The  same  conditions  apply  to  this  joint  as  for  Joint 
No.  2,  so  that  the  same  equations  will  have  to  be  used. 
In  the  equation  for  /,  Series  C,  the  factors  that  change  have 
the  following  values : 

^0  =  1153   square  feet,  H  =  $o  feet,  h  =  io  feet, 

lo  =  2 5   feet,  and  y0  =  ii.6  feet, 

and  by  substituting  them  in  the  equation,  completing  the 
square,  and  solving  for  /,  we  will  obtain 

£  =  29.5  feet. 
Using  the  equation  for  A  we  find  its  value  to  be 

A  =  1426  square  feet. 

Similarly,  using  the  equation  for  yt  with  the  new  values 
for  the  variables  as  indicated  above  we  have, 

=  i2  feet. 


86  HIGH    MASONRY    DAM    DESIGN 

JOINT  No.  3,  ICE  CONDITIONS 

Using  the  same  equation  here  that  was  used  for  Joint 
No.  2  (Stage  II  Series  D)  under  ice  conditions  with  the 
factors  taking  the  following  values:  ^0  =  1298  square 
feet,  Hi  =40  feet,  h  =  io  feet,  /0=35-5  feet,  and  ;yo  =  i3-3 
feet,  we  obtain  the  following  value  for  / : 

1=40  feet, 

while  the  use  of  the  same  factors  in  the  determination 
of  A  gives 

A  =  1676  square  feet, 

and  y  is  found  to  have  the  value  of 

y  =  14.5  square  feet. 

At  this  point  it  would  be  appropriate  to  determine 
whether  the  maximum  allowable  pressure  per  square  foot 
had  been  reached,  and  as  the  ice  conditions  for  Joint 
No.  3  are  more  severe  than  the  flood  conditions,  it  will 
be  applied  in  connection  with  the  former.  The  formula 
developed  in  the  theory  to  be  used  for  this  purpose  is,  since 

u  =— 
3 

2W7       3«\     2W       1676X7 
Maxf—  (*--)  =  —  =3Xl6X40  =  6.i  tons  per  sq.ft., 

which  shows  that  at  the  toe  of  the  dam,  50  feet  below 
the  top,  the  presence  of  ice  causes  a  pressure  well  within 
the  prescribed  limit  of  20  tons  per  square  foot. 


HIGH    MASONRY-  DAM    DESIGN  87 

JOINT  Nor  4,  FL<fbo  CONDITIONS 

Using  the  same  equations  (for  /,  Series  C,  Stage  II)  to 
determine    /,    A,   and   y,    but   with   the   following   values 
for  the  variable  factors,  A0  =  i426  square  feet,  H  =  6o  feet, 
h  =  io  feet,  /o  =  29.5    feet,  and  y0  =  12  feet,  we  find  that 
/=35  feet,     A  =1749  square  feet,     and    y  =  12.7  feet. 

JOINT  No.  4,  ICE  CONDITIONS 
The  same  equations  (Series  D  for  /)   will  be  employed 

here  as  in  determining  the  values  at  Joint   No.  3,   with 

the  following  values:  A)  =  1676  square  feet,   #1=50  feet, 

h  =  io  feet,  /o  =40  feet,  and  yo  =  14.5  feet. 

These  give  I  =45  feet,  ^=2101   square  feet,  ^  =  15.9 

feet,  and  p  =  6.S  tons  per  square  foot. 

JOINT  No.  5,  FLOOD  CONDITIONS 

With  the  same  equations  for  /,  (Series  C)  A,  y,  and  p, 
the  values  of  the  variables  in  which  have  become ;  A0  =  1 749 
square  feet,  H  =  jo  feet,  h  =  io  feet,  /0  =35  feet,  and  y0  =  12.7 
feet,  we  obtain  £=42.2  feet,  A  =2135  square  feet,  ^  =  13.9 
feet,  and  p  =  7.4  tons  per  square  foot. 

The  value  of  y  =  13.9  feet,  and  of  I  =42.2  feet,  establishes 

the  fact  that  at  this  point,  since  —  =  14.1  feet,  the  resultant 

\j 

pressure  for  reservoir  empty  has  passed  outside  the  middle 
third  by  0.2  foot.  It  would  seem  proper,  therefore,  to 
determine  the  value  of  q,  the  maximum  pressure  at  the 
heel,  to  see  if  the  limit  of  pressure  has  been  exceeded,  due 
to  this  excursion  of  the  resultant  beyond  the  middle  third, 
and  the  formula  to  be  employed  would  be 

2W 


88  HIGH    MASONRY   DAM    DESIGN 

which  is  a  modification  of 


Using  the  former  equation  and  substituting  the  appropriate 

values,  we  find 

q  =  7.46  tons  per  square  foot, 

which  is  well  within  the  limits  prescribed. 

JOINT  No.  5,  ICE  CONDITIONS 

Repeating  the  use  of  the  equations  (Stage  II  Series  D, 
for  /)  applied  in  solving  for  Joint  No.  4,  and  using  the  values  ; 
AQ  =  2101  square  feet,  HI  =60  feet,  h  =  io  feet,  /o  =45  feet, 
and  ^0  =  15-9  feet,  we  obtain  £  =  50.6  feet,  A  =2579  square 
feet,  y  =  17.5  feet,  and  p  =  7.4  tons  per  square  foot. 

It  should  be  noted  here  that  while  the  flood  conditions 
gave  a  value  to  y  for  Joint  No.  5,  which  showed  that  the 
resultant,  reservoir  empty,  fell  outside  the  middle  third, 
the  ice  conditions  indicate  that  the  resultant  lies  within 

the  middle  third  0.7  feet,  since  -  =  16.8  feet,   and  ^  =  17.5 

O 

feet.  Under  these  circumstances  then,  the  flood  conditions 
control,  and  make  it  necessary  to  batter  the  back,  while 
the  ice  conditions  do  not.  This  will  make  it  necessary  to 
use  the  equations  coming  under  Stage  III  for  the  flood 

conditions,   where   the  value   of   y=-   is   assigned,   while 

\) 

equations  under  Stage  II  will  be  used  for  ice  conditions. 

JOINT  No.  6,  FLOOD  CONDITIONS 

The  following  equation  under  Stage  III,  Series  C, 
must  be  employed  : 


HIGH    MASONRY    DAM    DESIGN  89 

t    » 

In  the  above  expression  the  factors  take  the  following 
values.  Note  that  h  is  here  made  20  feet.  ^0  =  2135 
square  feet,  H  =  90  feet,  h  =  2o  feet,  10=42.2  feet. 

Substituting  these  values  in  the  above,  completing  the 
square,  and  solving  for  /,  we  obtain 

/  =  66  feet. 

while  A  is  found  equal  to  3217  square  feet,  and 
Max.  p  =  7.i  tons  per  square  foot. 
It  is,  of  course,  unnecessary  to  determine  y  here,  since 

its  value  has  been  established  as  -. 

3 

It  is  necessary,  however,  to  determine  the  value  of 
/,  which  represents  the  amount  the  back  face  has  to  be 
battered  in  going  from  Joint  No.  5  to  Joint  No.  6.  Using 
the  following  equation,  in  which  all  the  quantities  are 
known,  we  have 


n0_      f 


JOINT  No.  6,  ICE  CONDITIONS 

We  will  again  use  the  equations  employed  in  solving 
lor  the  quantities  under  Joint  No.  5,  using  the  values 
^0  =  2579  square  feet,  H\  =80  feet,  h  =  2o  feet,  /0  =  5o.6 
feet,  and  ^0  =  17.5  feet. 

There  follows  from  these  values  1  =  62.4  feet,  A  =3709 
square  feet,  y  =  20.8  feet  and  p  =8.7  tons  per  square  foot. 


90  HIGH    MASONRY    DAM    DESIGN 

JOINT  No.  7,  FLOOD  CONDITIONS 

Using  the  same  equations  for  Joint  No.  7  that  were 
employed  for  Joint  No.  6,  but  increasing  the  value  of  h 
to  30  feet,  we  obtain  from  the  following  values  of  the 
factors,  A)  =32 1 7  square  feet,  H  =  i2o  feet,  h=^o  feet, 
/o  =  66  feet,  the  quantities,  £  =  92.5  feet,  ^=5595  square 
feet,  p  =  8.8  tons  per  square  foot  and  t  =  1.5  feet. 

JOINT  No.  7,  ICE  CONDITIONS 

It  will  be  noted  that  at  Joint  No.  6,  the  value  of 
y  =  20.8  feet  is  just  equal  to  -  =—  -  =  20.8  feet. 

O  O 

In  consequence  of  this,  it  will  be  necessary  to  take  a 
value  of  y  for  all  successive  joints  equal  to  -,  to  prevent 

O 

the  resultant,  reservoir  empty,  from  passing  beyond  the 
middle  third.  This  compels  the  use  of  Stage  III,  under 
Series  D,  which  will  produce  a  batter  in  the  back  face, 
and  the  following  equation  will  therefore  be  employed : 


Here  ^0=3709  square  feet,  HI  =  IIO  feet,  ^=30  feet, 
/o  =  62.4  feet,  which,  if  substituted  in  the  above,  give 

2  =  85.6  feet, 

while  A  =5929  square  feet,  p  =  io.i  tons  per  square  foot, 
and  t  =  2.o  feet. 

Down  to  Joint  No.  7,  two  cross-sections  have  now  been 
developed  side  by  side,  one  for  Flood  Conditions  and  one 


HIGH    MASONRY    DAM    DESIGN  91 

for  Ice  Pressure,  but- it  will  oe  noted  by  reference  to  the 
table  of  results,  and  to  the  profile,  that  to  and  including 
Joint  No.  5,  the  length  of  each  joint  under  Ice  Pressure 
has  exceeded  the  length  under  Flood  Conditions,  and  that 
at  Joint  No.  6,  the  difference  is  only  slightly  in  favor  of 
the  Flood  Condition  profile.  As  a  consequence,  down  to 
Joint  No.  6  the  section  that  must  be  used  is  that  estab- 
lished by  Ice  Pressure  formulae.  We  must,  therefore,  in 
examining  Joint  No.  7,  for  Flood  Conditions,  use  values 
for  area,  weight,  etc.,  which  represent  the  values  of  the 
Ice  Pressure  profile,  and  not  values  from  the  Flood  Level 
profile.  This  in  effect  amounts  to  investigating  Joint 
No.  7  and  its  successors,  whose  length  has  been  deter- 
mined by  Ice  Pressure  formulae,  by  the  equations  applying 
under  Flood  Level  Conditions,  and  where  the  latter  de- 
velops a  length  of  joint  in  excess  of  that  determined  by  Ice 
Pressure  the  greater  length  will  be  used. 

We  will  first  complete  the  cross-section  under  Ice 
Pressure  by  examining  Joints  No.  8,  9,  and  10,  each  of 
which  is  30  feet  below  the  next  above,  and  in  each  of  which 
the  same  formulae  apply  as  in  Joint  No.  7.  The  results 
alone  are  given. 

At  joint  No.    8,  Ice  Conditions  1=  108.6  ft.,  A  =    8,842  sq.  ft.,  £=11.9  tons,  t=  1.2  ft. 

9,    "  "  1=131.6  "    A  =12,445  "     "    £=13.8    "       /  =  o.8  " 

"       "       "    10,    "  "  1=155.1   "    A  =  i6,74S   "     "     £=I5.7    "       1  =  0.8" 

JOINT  No.    7,   FLOOD   CONDITIONS   COMBINED  WITH    ICE 
PRESSURE  PROFILE  ABOVE  JOINT  No.  6 

Proceeding  now  to  investigate  Joint  No.  7,  under 
Flood  Conditions,  with  the  Ice  Pressure  profile  above  Joint 
No.  6  providing  the  values  of  the  factors  to  be  used,  and 


92  HIGH    MASONRY   DAM    DESIGN 

employing  the  formulae  under  Stage  III,  Series  C,  we 
find,  using  A0=370Q  square  feet,  H  =  i2o  feet,  h=$o  feet, 
and  £0  =  62.4  feet, 

/  =  82.p  feet,        A  =5889  square  feet, 
p  =  10.3  tons  per  square  foot         and         t  =  1.2  ft. 

But  a  comparison  of  these  results  with  those  obtained 
for  the  same  joint  under  Ice  Pressure  shows  that  the 
latter,  being  larger,  controls  the  length  of  joint. 

JOINT  No.   8,   FLOOD   CONDITIONS    COMBINED  WITH    ICE 
PRESSURE  PROFILE  ABOVE  JOINT  No.  7 

Similarly,  for  Joint  No.  8,  we  must  employ  the  factors 
resulting  from  the  Ice  Pressure  profile  in  the  solution  of 
the  quantities  in  the  Flood ,  Conditions.  These  factors 
become  ^0  =  5929  square  feet,  H  =  i$o  feet,  ^=30  feet, 
and  /0=85.6  feet. 

From  which  we  derive,  by  the  use  of  the  same  formulae 
employed  under  Joint  No.  7,  the  following:  I  =  111.2  feet, 
.A  =888 1  square  feet,  p  =  n.6  tons  per  square  foot,  and 
2  =  1.9  feet. 

Here  we  find  the  above  quantities  exceeding  in  value 
those  determined  by  the  Ice  Pressure  formulae,  so  that  the 
former  must  be  employed.  In  other  words,  /  =  111.2  feet 
is  used  in  the  profile  instead  of  /  =  108.6. 

JOINT   No.    9,    FLOOD   CONDITIONS    WITH    ICE    PRESSURE 
PROFILE  COMBINED 

The  factors  to  be  used  are  those  just  derived,  A  o  =8881 
square  feet,  H  =  iSo  feet,  h=^o  feet,  and  IQ  =  111.2  feet. 


HIGH    MASONRY    DAM    DESIGN 


93 


From  which  we  obtain  witft  the  same  formulae,  £  =  138 
feet,  A  =12, 6 1 9  square  feet,  £  =  13.3  tons  per  square  foot, 
and  t  =  i.6. 

Here  again  the  values  of  the  above  quantities  are 
greater  than  those  determined  by  Ice  Pressure  formulae, 
so  they  must  be  used. 

JOINT  No.   10,  FLOOD  CONDITIONS    WITH  ICE    PRESSURE 
PROFILE  COMBINED 

The  factors  are  ^0  =  12,619  square  feet,  H  =  22o  feet, 
h=3o  feet,  and  lQ  =  i$8  feet;  whence,  /  =  161.9  feet,  A  — 
17,119  square  feet,  £  =  15.4  tons  per  square  foot,  and 
t  =0.3  foot,  which  values  are  again  in  excess  of  those  obtained 
from  the  formulae  for  Ice  Pressure. 

In  order  to  check  the  above  calculations  by  the  graph- 
ical method,  there  is  submitted  in  tabular  form 

QUANTITES   FOR   GRAPHIC    SOLUTION 


Water 
Units. 

Water  Units 
in  Tons.* 

w                                  = 

I7.II9X-       = 

2I02 
2 
I62X2IO       2 

4O,OOO 
22,050 
H,340 
752 

1250 
705 
354 
23-5 
1130 

Water  Pressure  = 

Ice  Pressure                                   — 



2                3 

47,000 

Resultant     . 

62.5 

*  Multiply  62.5  into  Water  Units  to  get  pounds,  or  by  ,03125  for  tons. 


94 


HIGH    MASONRY    DAM    DESIGN 


"8 


f.    -2 


w    5 


itM>ONN>O>-i<N'<t>-iNi-i<N(>ot^Tj-r<:i-ivOO  t-  roo 


O  O  Oi/3>ow  OO  ^O  t^    .  Tt    . 


M       -OOIH       -Ol- 


•00  O     •  w      •  fO     •  >0        2        O  M  C0i 

.  M.M.M-M          p  MMM, 


O  O  O  O  O 


lO    •  OJ     •  O*    •  O       "S        to  M  o  O 


O  f5  O  >O  t*  O>  O\  lO  OOO 


O  IO  t^  O\  O\  >O  O  00  00 


p^MMNMro^rJ-Niorot^iNOOoO     <VO     -fO    -M        fl       oo  t^O  ' 


OOOOOOlOiOOOONvOOTt-i 
W  w  c^  C^  ro  CS  fO  ! 


1-1  (~0 


^  O  O  O  O  1010O  O  O  NO  O  -ttiOO     -O-O-O  ^  O<NOO 

fC<N  (*)<N  trOTj-TtioOO  aoo     -O     -co    -in  g  OO  M  roo 

. ^^o^oooo  o.M^a^o.^a..  1  ^T^T 

jO1      M"  M"  M  M  M  M  M  w  N  N  ro  ro  10  10    ;  oo     ;  (N     ;  o"  6  1000"  ri  t^ 


.     .  to  if)  O  rooo 
.     .iHPO.rO>OO\ 

.       .  >O    O    W    M    N 


:°°  ;  t   ^      :  °J00.V°. 
;oo    ;  N      JJ       ]  1000  N 


-  1^00  00  O  O  CO  fOO  \O  Oi  O\ 


°°  s 


OOOOOOOOOOOOOOOO   -5   OOOl 

>  IOOO  ^•ThMMMMMMC^CNrOfOfOf'OfOfOfOCO       te        rOfOfOi 

«    TtM  g 


HIGH    MASONRY    DAM    DESIGN 


95 


The  finally  adopted  cross-sections  of  the  Olive  Bridge 
Dam  and  the  Kensico  Dam,  in  addition  to  being  investi- 
gated by  means  of  the  formulae  of  Chapter  IV,  and  others 


El. 610 


"Ice  Pressure"  Condition 

"Flood" 

"     with  "Ice"    " 

Figures  refer  to  combination  of  Ice 
Section  above  joint  Xo.  7  with  Flood 
Design  below 

Limits  of  middle  thirds  marked  with, 
small  circles  thusio 


00) 


54  54 

CALCULATED   CROSS-SECTION 


E1.39C 


NOTE. — Earth  on  down-stream  toe  would  not  increase  compressive  stress  beyond  20  T.  per 
square  foot,  if  taken  at  100  pounds  per  cubic  foot. 

FIG.  16. 

like  them,  taking  into  account  variation  in  distribution 
of  uplift,  were  subjected  to  studies  based  on  the  method 
of  Ottley  and  Brightmore,  given  in  Chapter  VIII,  for 
determining  the  shearing  stresses  to  be  expected  on  vertical 


96 


HIGH    MASONRY    DAM    DESIGN 


planes.  This  maximum  shear  intensity  was  foundi  to  occur 
within  8  or  9  feet  of  the  down-stream  edge  of  the  base, 
in  each  case  and  did  not  exceed  92  pounds  per  square 


Inspection 

Guile 
Flow  Li 


Concrete  drainage  > 
blocks 


El  .500 
Drainage  Well 


FIG.  17.— Olive  Bridge  Dam. 

inch.  This  was  with  the  assumption  that  the  base  (at 
the  foot  of  foundation  excavation)  made  an  acute  angle 
with  the  down-stream  face.  The  shear,  of  course,  was 


HIGH    MASONRY    DAM    DESIGN 


97 


zero  at  the  up-stream  and  down-stream  edges  of  the  base. 
Incidentally,  the  assumed  masonry  density  was  checked 
with  the  final  design,  allowing  for  all  openings  within  the 
structure. 

Besides  these  investigations  special  studies  were  under- 


CROSS-SECTION  AT  TOP 
2. 0  .    8       ±        68       10ft. 


taken  to  ascertain  the  probable  effect  on  the  stresses,  of 
the  large,  stream  flow  passageway  in  the  lower  part  of  the 
Olive  Bridge  Dam  left  open  during  construction  and  sub- 
sequently closed  with  concrete;  also  the  effect,  in  the  case 
of  the  Kensico  Dam,  of  the  lower  portion's  cracking  longi- 
tudinally, due  to  temperature  changes  during  construe- 


98  HIGH   MASONRY   DAM   DESIGN 

tion.  This  cracking  was  assumed  to  take  place  so  as  to 
divide  the  lower  portion  into  three  parts  that  would  serve 
as  three  lines  of  huge,  supporting  blocks  for  the  portion 
above.  The  stresses  that  would  probably  result  in  each 
case  were  found  to  give  no  cause  for  concern. 


CHAPTER  VI 
WEIR  OR  OVERFALL  TYPE  OF  DAM 

IN  the  following  pages  the  development  of  the  cross- 
section  for  the  overfall,  spillway,  or  weir  type  of  dam  will 
be  considered,  the  function  of  which  type  is  to  permit  the 
flow  of  water  over  the  top,  or  "  crest."  This  class  of 
structure  may  serve  primarily  either  of  two  purposes : 

(1)  To  make  accurate  measurements  of  the  discharge 
over    the    crest,   the   weir   being,   in  that  case,   called    a 
1 '  measuring  weir. ' ' 

(2)  To  allow  surplus  water  to  escape  from  the  side 
of  a  canal  or  reservoir,  the  weir  being  then  styled  a  "  waste 
weir  "  or  "  spillway." 

The  same  structure  may,  however,  serve  the  double 
purpose  at  one  and  the  same  time,  as,  for  example,  where 
the  flow  over  a  spillway  dam  is  gaged. 

In  the  first  case,  the  discharge  capacity  per  unit  length 
of  weir  crest  is  known  in  terms  of  the  depth  or  head  of 
water  on  the  crest,  while  in  the  second  case,  the  discharge, 
usually  a  maximum,  is  either  known  or  assumed,  and  the 
length  of  weir  necessary  for  a  given  head  or  allowable  range 
in  head  is  thereby  determined. 

Among  the  many  proposed,  the  simplest  form 
of  expression  for  weir  discharge  is  the  Francis  formula, 

99 


100  HIGH    MASONRY   DAM    DESIGN 

which   shows   the   relation   among   those   factors   entering 
into  the  evaluation  of  the  discharge,  as  follows : 

Q  =  CLH*1*, 
in  which 

Q  —  volume  of  discharge  per  unit  of  time ; 

C  =  an  empirical  coefficient ; 

L=  length  of  weir  (corrected  for  end  contractions,  if  any 

of  the  issuing  sheet) ; 
jy=head  on  the  crest  corrected  for  the  effect  of  velocity 

of  approach. 

It  is  unnecessary  to  give  here  more  than  passing  con- 
sideration to  the  subject  of  the  hydraulics  of  weirs  in  its 
various  phases;  the  reader  is  referred  for  fuller  discussion 
to  almost  any  work  on  hydraulics,  but  especially  to  weir 
experimentation.  * 

As  it  is  impossible  to  predetermine  exactly  the  dis- 
charge that  is  to  pass  over  a  waste  weir  and  as  allowance 
must  always  be  made  for  unusual  storms  or  floods  in 
providing  a  length  of  crest  in  any  given  case,  a  knowledge 
of  the  precise  discharge  capacity  of  the  waste  weir  will 
be  of  less  importance  than  in  the  case  of  a  measuring  weir. 

Nevertheless,  in  addition  to  the  question  of  spillway 
length  to  be  provided,  there  remains  the  problem  of  arriving 
at  a  proper  form  of  cross-section  for  the  structure.  This 
latter  may  be  determined  from  a  knowledge  of  the  dis- 
charge, even  though  it  be  inexact,  since  it  leads  to  approx- 

*  Merriman's  "  Treatise  on  Hydraulics;"  Traut wine's  "  Engineers'  Pock- 
etbook;"  "  Water  Supply  and  Irrigation,  Paper  200"  on  "  Weir  Experi- 
ments, Coefficients  and  Formulas,"  Dept.  of  Interior,  U.  S.  Geol.  Survey; 
"  Hydraulics  of  Rivers,  Weirs  and  Sluices,"  by  David  A.  Molitor  . 


HIGH    MASONRY   DAM    DBSIQlf  *->•;•-',  j^     \*  iOl 

imate  probable  velocities  attained  by  the  sheet  of  water 
in  its  fall  over  the  crest  together  with  the  shape  of  the 
sheet. 

One  condition  to  be  fulfilled  by  a  spillway  dam  is  that 
it  shall  discharge  a  maximum  quantity  of  water  per  unit  of 
time  for  a  given  length  and  head  on  crest,  and  without 
endangering  the  structure  in  any  way.  It  is  evident  that 
for  high  heads  and  consequent  heavy  sheets  of  water, 
the  best  results  will  obtain  with  regard  to  the  structure's 
safety,  if  the  overfall  takes  place  without  shock  at  any  point ; 
that  is,  if  the  structure  is  fitted  smoothly  to  the  shape 
that  the  sheet  would  naturally  assume  in  flowing  over 
the  crest,  and  if  the  cross-section  at  the  bottom,  down- 
stream, is  of  such  form  and  material  as  to  lead  the  water 
away  from  the  structure,  without  impact  or  erosion  at  the 
toe. 

It  has  been  observed  that  a  discharge  over  a  spillway 
dam  has  produced  vibrations  that  may  affect  the  stability 
of  the  structure.  These  may  be  accounted  for  as  follows : 

If  the  sheet  of  falling  water  leaves  the  dam's  face 
and  then  impinges  upon  it  lower  down,  air  will  be  entrained 
in  the  intervening  space  that  will  be  gradually  exhausted 
by  the  rapidly  moving  filaments  of  the  adjacent  sheet. 
As  the  condition  of  a  vacuum  is  approached  the  superior 
atmospheric  pressure  deflects  the  entire  sheet  of  water 
violently  against  the  face  of  the  dam,  causing  a  shock, 
provided  the  mass  of  falling  water  is  not  too  great  to 
resist  such  deflection.  The  atmospheric  pressure,  acting 
also  on  the  masonry  of  the  dam,  will  tend  to  force  it  down- 
stream, if  not  of  sufficient  mass,  during  maintenance  of 
a  vacuum  under  the  sheet.  This  is  the  so-called  "  suction  " 


»»  •^^»**j»^*S*     i   o.- 

1-Q2  HIGH    MASONRY   DAM    DESIGN 

down-stream  exerted  on  the  masonry.  Repeating  cycles  of 
first  entraining  air,  then  exhaustion,  and  then  readjust- 
ment, coupled  with  the  consequent  changes  in  the  imping- 
ing sheet  lower  down  the  face,  produce  the  vibrations 
above  referred  to. 

The  danger  from  these  lies  in  the  possibility  of  action 
in  an  up  and  down-stream  direction  and  of  acting  sym- 
pathetically with  the  structure's  rate  of  vibration,  in 
which  case  the  cumulative  effect  might  lead  to  the  ultimate 
destruction  of  the  dam. 

The  smooth  face  fitted  to  the  curve  of  fall  would 
prevent  the  water  suddenly  leaving  it  at  any  point.  Fur- 
thermore, the  insurance  against  the  formation  of  a  vacuum 
between  the  face  of  the  dam  and  the  water  sheet  may 
be  accomplished  by  so  proportioning  the  face  of  the  dam 
that  the  cross-section  would  extend  well  into  the  water 
sheet  throughout  its  entire  extent. 

Fanning  *  recommended  a  down-stream  face  of  the 
weir,  "  slightly  more  full  than  the  parabolic  curve  which 
the  film  of  water  at  two-thirds  depth  on  the  crest  tends 
to  take,"  in  order  that  the  overflowing  water  might  not 
lose  contact  with  the  masonry  at  any  point.  The  foot 
of  this  curve  should  be  joined  with  the  river  bed  by  a 
vertical  curve  of  approximately  100  feet  radius,  f  (Fanning) 
tangent  to  the  face  curve  and  to  the  river  bed.  He  further 
indicated  how  the  above  face  curve,  thus  forming  what 
is  known  as  the  ogee  cross-section  for  high  weirs,  may  be 

*  "  A  Treatise  on  Hydraulic  and  Water  Supply  Engineering,"  Ed.  1899, 
by  J.  T.  Fanning. 

t  For  high  dams.  This  radius  would  be  far  too  high  a  value  for  moderately 
high  dams.  The  slope  of  the  face  and  valley  downstream  would  regulate 
the  radius  at  toe  to  a  great  extent. 


HIGH    MASONRY    DAM    DESIGN 


103 


resolved  into  steps,  for  the  purpose  of  breaking  the  fall  of 
water  into  a  number  of  smaller  falls.  The  steps  "  kill  " 
acceleration  of  the  falling  sheet.  It  is  obvious  that  the 
force  of  the  falling  water  in  such  modified  section  is  con- 
strained to  act  in  a  vertical  direction.  The  "  steps  " 


FIG.  19. 

should  be  so  proportioned  that  they  will  project  well  into 
the  under  surface  of  the  sheet.  Figures  19  and  20  illus- 
trate, respectively,  these  ogee  and  stepped  types  and  are 
cross-sections  of  recent  structures  built  for  heads  not  much 
over  5  feet.  It  may  be  noted  that  the  tops  are  smooth 
curves  in  both  types  of  section,  that  is,  stepping  should 
not  occur  until  the  sheet  is  well  over  the  crest.  The 
stepped  type  serves  where  high  velocities  at  the  down- 


104 


HIGH    MASONRY    DAM    DESIGN 


stream  toe  are  objectionable.  In  the  matter  of  construction, 
face  stones  should  be  set  on  edge  radially  with  respect  to 
the  curves  of  the  face,  especially  at  the  foot.  In  a  con- 
crete structure,  hard  stone  is  often  thus  embedded  at  this 
lower  portion  to  resist  erosion. 


A  parabola  can  be  determined  to  fit  the  shape  of  the 
falling  sheet  and  extend  just  inside  its  lower  surface,  or 
nappe,  as  the  French  term  it. 

First.  General  formulae  for  design  Eqs.  (2)  and  (3)  and 
Eqs.  (9)  and  (10),  will  be  derived,  as  has  been  done  for 
the  gravity  dam,  in  Chapter  III,  and 

Second.  Consideration  will  be  given  to  the  shape  of 
the  nappes  and  the  determination  of  a  value  for  the  par- 
ameter of  the  parabola,  appearing  in  the  general  formulae 
as  a  factor. 


HIGH  MASONRY^DAM  DESIGN  105 

FORMULA  #OR  DESIGN. 
Spillway  Dam. 

Derivation. — In  Series  F,  Formulae  for  Design,  Stage 
II,  page  67,  there  appears  an  expression  by  which  a  trap- 
ezoidal cross-section  for  a  spillway  dam  may  be  calcu- 
lated. From  the  preceding  discussion  it  is  evident  that  the 
manner  of  determining  the  shape  at  or  near  the  top  re- 
quires a  more  detailed  treatment ;  but  the  formulae  of  Series 
F  are  sufficient  for  fixing  the  cross-section  lower  down. 

One  method  would  be  to  round  the  top  of  the  trape- 
zoidal section  in  a  practical  way  to  suit  the  case  in  hand 
after  comparison  with  crests  of  approved  existing  struc- 
tures, knowing  their  discharge  capabilities.  The  structures 
of  Figs.  19  and  20  were  designed  according  to  this  method. 

In  the  following,  however,  a  general  parabolic  section 
is  derived,  corresponding  to  Stage  I,  or  the  rectangular 
section,  of  the  series  of  formulae  for  design  heretofore 
given  and  following  the  identical  principles. 

As  the  shape  of  the  parabolic  section  is  fixed  by  the 
falling  sheet  of  water,  there  being  no  possibility  of  ice 
thrust  during  overfall,  and  as  ice  thrust  near  the  top, 
with  no  overfall,  would  have  to  be  resisted  by  proper 
reinforcement,  vertically,  near  the  up-stream  face,  it  would 
seem  reasonable  to  ignore  that  feature  in  the  formulae 
for  design.  But,  as  the  extent  downwards  from  the  crest 
of  the  parabolic  section  would  be  affected  by  such  thrust 
near  the  crest,  it  is  thought  desirable  to  include  this  factor, 
for  the  purpose  of  investigating  its  effect,  if  for  no  other 
reason.  Formulae,  Eqs.  (9)  anol  (10),  containing  the 
factor  T,  with  water  surface  at  or  below  the  crest,  will 


106 


HIGH    MASONRY    DAM    DESIGN 


be  given,  therefore,  immediately  after  those  first  derived 
for  overfall  conditions. 


____rx 
heoretical  Crestx 


FIG.  21. 

As  before,  the  middle  third  limit,  for  center  of  result- 
ant pressure  on  any  horizontal  joint,  will  be  retained. 

Likewise,  for  any  given  case,  the  terms  or  factors 
expressing  other  than  the  imposed  conditions  of  loading 
must  be  equated  to  zero. 


HIGH    MASON»RY    DAM    DESIGN  107 

'> 

Cases  may  arise  where  the  highest  head  of  water  over- 
topping the  dam  does  not  provide  the  critical  stage  for 
stability,  due  to  a  rapidly  rising  backwater  with  the  in- 
creased discharge.  Such  must  be  especially  investigated 
for  determination  of  the  critical  head. 

In  addition  to  the  nomenclature  given  in  Chapter 
III,  p.  31,  et.  seq.,  the  following  designations,  in  connection 
with  Fig.  2 1 ,  will  be  employed : 

b'  =  vertical  distance  from  the  water  surface  in  the  res- 
ervoir down  to  the  theoretical  crest  of  spillway 
when  the  crest  is  overtopped. 

kb' =  vertical  distance  from  the  water  surface  down  to 
the  actual  crest  of  spillway  when  overtopped. 
(kb'=b,  of  Chap.  III.) 

&=o.888  (Bazin). 

k'b'  =  horizontal  distance  between  the  vertical  line  through 
the  crest  (crest  line)  and  the  up-stream  vertical 
face. 

£'=0.25  (Bazin). 

X  =  any  abscissa  (vertical)  of  the  down-stream  face  of 
spillway. 

y  =  ordinate   (horizontal),   corresponding  to  X,  origin  of 

rectangular  co-ordinates  being  at  the  actual  crest. 
Y2=Kb'X,  equation  of  the  parabolic,  down-stream  face 
of  the  spillway  cross-section,  with  respect  to  the 
vertical,  X  axis  and  the  horizontal,  Y  axis  through 
the  actual  crest.  (Kb '  is  the  parameter  of  the 
parabola.) 

K  =  the  constant  (considered  later  in  this  chapter)  de- 
termining the  parabolic  face,  above,  so  that  the 


108  HIGH    MASONRY   DAM    DESIGN 

curve    shall    extend    within    the    falling    sheet    of 

water. 

7^  =  2.25  has  been  used.* 

y=the  ordinate  to  the  centroid  of  the  parabolic  seg- 
ment of  the  cross-section  above  any  joint  at  dis- 
tance X  below  the  origin. 
y'-fy. 

Area  of  segment  =|XF,  down  to  any  level  X  below 
the  origin. 

Overfall,  or  Flood  Level  Design  (No  Ice). 
In  connection  with  the  following,  see  Fig.  2  1  . 
Overturning  moment  due  to  : 

(a)  Horizontal  static  water  pressure  on  back  (head  =h\). 

(b)  Upward  water  pressure  on  base;   pressure  intensity 
decreasing  uniformly  from  cHf  01   c(/&i+fe)r,  at  heel  to 
zero  intensity  at  toe. 

(c)  Mud  (liquid)  pressure  on  back  (head  hz)  as  before. 

(d)  Dynamic  pressure  of  water,  DJ-. 

(e)  Water  flowing   over   top   of  dam,   weight   of   water 
of  depth  b,  on  top  of  dam  being  neglected. 

For  condition  of  no  dynamic  pressure,  D  =o. 

For  condition  of  no  upward  water  pressure,  c  =  o. 

For  condition  of  no  mud  (i.e.,  mud  being  replaced  by 
water)  make  h%  =o,  hi  =  H. 

As  in  the  previous  deductions,  the  fundamental  equa- 
tion for  length  of  joint,  /,  is 


M 

or  l  =  u+— 


*  Mr.  Richard  Muller  in  "  Engineering  Record  "  of  Oct.  24,  1908. 


HIGH    MASONRY    DAM    DESIGN  109 

«  t 

in  which  ' "•-**' 


2A 


L         6A  6A 

In  deducing  the  expression  for  A,  or  the  total  area 
of  the  cross-section  above  the  joint  /,  the  top  will  be  con- 
sidered horizontal  up-stream  from  the  crest  line. 


A  =  krb\hi  +h2-kb')  +f  (Jn  +h2  -W)  \fKV(h 
or 


u=-     and     l  =  k'br- 
3 

(This  expression  for  /  may  be  more  conveniently  held 
for  the  last  substitution  in  derivation.) 

y  may  be  derived  by  taking  static  moments  about  the 
up-stream  edge  of  the  joint  /,  as  follows : 

-kb')+^(hl+h2-W)(l-kfbf)(l(l-kfbf)+kfbf} 


k'b'(hi+h2-kbf)+%(hi+h2-kb')(l-k'b') 
whence 


_     3 

''   4    +4(k' 

M 

Substituting  the  above  values  for  a,  y,  —  and  A  in 

the  fundamental  equation  for  /,  and  reducing,  gives 


110  HIGH    MASONRY    DAM    DESIGN 


General  Formula: 

By  substituting  in  this  last  expression  values  of  /  and 
I2  involving  K,  as  suggested  above,  and  reducing,  gives 

-  6/ii3  +  (hi  +/*2)2(7  A  -  6c)Kb' 


+  (hi  +fo,)  1  2k'b'[(8  A  -  6c)  VKb'(hi  +h2-  W) 


+  iSDh2  +kb'[(6k'b')2A  -k(b')2(7KA  -i2k)-  iSD]     .     (i) 


From  this  General  Formula,  Eq.  (i),  there  may  be 
derived  directly  Eqs.  (2)  and  (3),  formulae  for  the  parabolic 
cross-section  corresponding  to  those  for  the  rectangular 
cross-section  presented  in  "  Series  F,  Stage  I,  Cases  (i) 
and  (2)  "  of  Chapter  III,  p.  66. 

Case  (i).  Condition  hi=H;  h2=o,  or  no  mud,  H, 
to  be  determined,  underlined  in  Eq.  (2). 


+H\2k'b'[(6c  -  8A)  VKb'(H  -  kb')  +3  (c  -  A)fc' 


+  i6kk\b')2&VKb'(H_-kb')=kb'[iSD-6(k'b')2A 

....     .    ,     .  (2) 


HIGH    MASONRY   DAM    DESIGN  111 

Case    (2).     Condition   hi   of  known  value,   fe,   to   be 
determined,  underlined  in  Eq.  (3). 


'b'[(6c  - 8 A)  ^/Kbf(hz  +hi  -kb) 
+3  (c  -  A)fc  V]  -  2k(b')2[(3c  -  7  A)  K  +9k) 


-6h1*,    .     .     (3) 

It  will  be  noted  that  the  unknowns,  in  Eqs.  (2)  and  (3) 
above,  are  H  and  &2,  respectively,  and  that  the  known 
terms  or  factors  are  recurrent,  and  in  any  design  need  be 
substituted  but  once. 

A  convenient  method  of  using  either  of  the  above  Eqs. 
(2)  or  (3)  is  as  follows: 

(1)  Substitute    chosen  values    for    the    known  terms, 
according  to  assumptions  and  conditions,  and  reduce.    The 
right-hand  member  of  either  of  the  equations  reduces  to  a 
number,  the  left-hand  member  (an  expression  of  the  third 
degree)  to   terms  involving  the  unknown  (H  or  h*)  and 
numerical  factors. 

(2)  Substitute  at  least  three  successive  trial  values  for 
the  unknown  (these  values  always  positive)  and  compute 
the    corresponding    values   of    the    left-hand  side   of  the 
equation. 

(3)  Plot  a  curve  with  these  trial  values  for  the  unknown 
as  ordinates  and  the  values  of  the  left-hand  member  cor- 
responding  as   abscissae.     The   ordinate   corresponding   to 
the    predetermined  value  of    the  right-hand    member  will 
at  once  yield  the  correct  value  of  the  unknown. 


112  HIGH    MASONRY    DAM    DESIGN 

Illustration  of  Use  of  Equations.  Condition. — Hydrostatic 
water  pressure  on  back,  with  overtopping.  Eliminating  all 
but  hydrostatic  and  overtopping  factors  in  Eq.  (2),  above, 
gives,  as  a  result,  Eq.  (4),  following: 


6H*  -  >/KbfH2A  -H\2k'b'A[sVKb'(H-kb' 


] (4) 

Eq.  (4)  will  serve  to  illustrate  the  foregoing  remarks, 
using  the  numerical  coefficients  for  k,  kf  and  K,  respectively, 
0.89,  say;  0.25;  and  2.25.  Substituting  these  values  in 
Eq.  (4)  and  reducing,  gives  an  expression  in  terms  of  b', 
A,  and  H,  the  last,  the  unknown  to  be  determined,  as 
follows : 


[%V2. 25b'(H-o.89b') 
+0.06256']  -0')2(4.6725A  -2.3763)! 


-0.896') 
- 1.4099). (5) 

Eq.  (5),  then,  contains  Bazin's  coefficients  *  and  a 
constant,  fixing  the  parabolic  face  that  will  determine  a 
cross-section  presumably  acceptable  as  to  flow  conditions. 
The  conditions  of  stability  down  to  the  depth  of  water,  H, 
on  the  base  at  that  depth,  is  that  of  the  "  middle  third 
limit,"  for  the  resultant  pressures  on  the  joints  at  and 
above  that  base.  The  loading  conditions,  for  simplicity, 
comprise  only  those  of  horizontal  static  water  pressure 
on  back  and  "overtopping." 

*  See  Table  III. 


HIGH    MASONRY    DAM    DESIGN 


113 


Assuming  a  head  (&')  on  the  theoretical  crest  of  20 
feet  and  A  =  2.  2  4  and  inserting  these  values  in  Eq.  (5) 
gives 


+  53i.6.23V45(#-i7.8)  =24,984.     (6) 

Preliminary  to  the  prosecution  of  a  solution  of  Eq. 
(6),  it  will  prove  convenient  to  calculate  and  plot  a  curve 
(or  the  value, 


—  17.8),  for  different  values  of  H, 
ranging  from  18  to  100  in  this  case.  The  smooth  curve 
resulting  can  be  used  in  the  subsequent  assumptions  for 
''trial"  H  's,  to  pick  off  corresponding  values  of 


—  17.8) 
for  entry  in  the  left-hand  member  of  Eq.  (6).     This  factor, 


—  17.8),  is  useful  in  indicating  at  once  the  lowest 
positive  value  that  may  be  assumed  for  H.  In  the  case 
here  given  H  cannot  have  a  positive  value  less  than  17.8. 

Without  reproducing  the  curves,  the  results  of  the 
trial  values  of  H,  assumed,  together  with  the  correspond- 
ing values  of 


—  17.8)  obtained  from  the  prelim- 
inary curve  are  contained  in  the  following  table,  for  the 
case  here  under  consideration  : 


Assumed 
H. 

Left-hand  Member 
of  Eq.  (6). 

V4S(H_i7.8). 

95 

58.8 

-40,325 

IOO 

60.8 

-11,678 

105 

62.6 

+27425 

From  the  curve  (plotted  with  the  assumed  values  of 
H  in  the  first  column,  and  the  corresponding  numbers  of 


114  HIGH    MASONRY    DAM    DESIGN 

the  third  column  of  the  table,  as  abscissae  and  ordinates, 
respectively)  the  ordinate,  of  value  +24,984,  of  Eq.  (6), 
gave  as  its  abscissa,  a  value  of  H  equal  to  104.7  ^eet-  That 
is,  this  dam  may  be  extended  as  a  "  parabolic  section  " 
down,  until  the  head  of  water  on  its  base  becomes  104.7 
feet,  with  a  head  of  20  feet  on  the  theoretical  crest.  The 
distance  below  the  actual  crest  would  be  #—&£/  =  104.  7 
-17.8=86.9  feet. 

(It  may  be  said  at  this  point  that  an  investigation  for 
positive  roots  of  Eq.  (6),  between  18  and  25,  for  values  of 
H,  results,  for  the  left-hand  member  of  that  equation, 
in  decreasing  values,  as  H  varies  from  18  to  25;  below 
1  8,  the  root  would  have  to  be  negative. 

The  factor  V(H  —  17.8)45  indicates  an  imaginary  quan- 
tity at  this  stage.) 

A  dam  of  the  above  type  and  subjected  to  the  assumed 
conditions,  would  therefore  reach  the  limit  of  stability 
contained  in  the  expression  "  the  middle  third  limit," 
at  nearly  87  feet  from  its  crest. 

This  result  may  be  checked  by  use  of  the  formula  for 
investigation,  i.e.,  calculate  the  position  on  the  base  of 
the  center  of  pressure  for  20  feet  head  on  the  theoretical 
crest  and  base  of  parabolic  section  84.7  feet  below  that 
crest.  Making  the  proper  eliminations  and  substitutions 
to  suit  the  parabolic  section  in  the  first  formula  for  in- 
vestigation of  Chapter  IV,  p.  70,  there  results: 


(6a) 


. 

o./i  A 

The  area  A  may  be  expressed  in  terms  of  /,  as  follows  : 
A=$(H-kb')(k'b'+2l).     .  (66) 


HIGH    MASONRY    DAM   DESIGN  115 

t 

Whence,  from  (6a)  and  $6)  ,  there  is  obtained  : 


y  - 

''  ' 


Substituting  in  Eq.  (6c)  the  values  : 
(H-  kb')=S6.  9  feet; 


l=k'b'+VKb'(H-kbf)  =67.5  feet; 
H  =  104.7  feet; 
:  A  =  2. 24; 

y= — +-/z7/L/-L~n=25-66feet; 

4       4(k  b  +  2L) 
I—  y  =41.84  feet, 

there  results,  after  reduction, 

u  =41.8  —  19.4  =  22.4  feet. 

But  -  =  22.5  feet,  which  is  a  sufficiently  close  agree- 
o 

ment. 

To  continue  the  cross-section  below  a  head  of  104.7 
feet,  or  the  foot  of  the  parabolic  section,  recourse  must 
be  had  to  Formulae  of  Design,  Series  F,  Stage  II,  (b), 
or,  in  case  an  ice  pressure  cross-section  is  being  calculated, 
similar  formulae  of  Series  E,  Stage  II,  et  seq.,  apply. 

Ice  Pressure  Design. 

The  formulae  now  to  be  derived  are  similar  to  those 
of  Series  E,  Stage  I,  of  Chapter  III,  with  the  same  con- 
ditions governing  for  that  series,  as  to  loading,  that  is: 

(a)  Horizontal,  static  water  pressure  on  back,  (head  hi) ; 

(b)  Ice  pressure  at  surface  of  water,  (Tf) ; 


116  HIGH    MASONRY   DAM   DESIGN 

(c)  Upward  water  pressure  on  base  ; 

(d)  Mud  (liquid)  pressure  on  back  (head  hz), 
commencing  at  distance  h%  above  joint  in  question. 

In  the  fundamental  formula  for  /,  M  here  differs  from 
the  value  given  in  the  preceding  consideration,  in  that 
the  overtopping  effect  and,  consequently,  dynamic  pres- 
sure, are  here  supplanted  by  the  effect  of  the  ice  pressure, 
which  will  be  referred  to  the  top  of  the  dam,  in  location, 
by  the  vertical  distance  a\.  This  reference  will  in  turn 
change  the  expressions  for  A  and  /,  of  the  parabolic  section. 

The  procedure,  otherwise,  remains  the  same  as  for  the 
overfall  condition. 

There  follow: 


M 


A  7' 

Substituting  these  in  l=u-\  —  j-+?>  and  reducing,  gives: 


(7) 


The  equation,  l  =  k'b'  +  VKb'(hi+h2+ai),  applies  here. 
Substituting  this  expression  for  /,  together  with  its  square 
in  Eq.  (7)  and  reducing,  gives  Eq.  (8),  whence  two  cases, 


HIGH    MASONRY    DAM    DESIGN  117 

previously  formulated  for  the  overflow  condition,  follow 
directly. 

Eq.  (8)  is: 


i-\-h2 
+  7Kb'(hi  +h2  +ai)]  }  =-^l  .......     (8) 

From  Eq.  (8)  may  be  obtained  : 

Case  (i).  Condition,  hi=Hi\    h2=o,  or  no  mud.     Hi 
to  be  determined. 


(9) 


Case  (2).  Condition,  hi  of  known  value,  h2  to  be  de- 
termined : 

^+(fe+&o 


+6c(k'b'  +  k'V  VKb' 


h2-\-ni 

-6hi*.    ....     ....     (10) 


118  HIGH    MASONRY   DAM  .DESIGN 

To  recapitulate : 

Eqs.  (2)  and  (3)  of  this  chapter  are  the  working  equa- 
tions for  the  overfall  design;  Eqs.  (9)  and  (10)  for  the  ice- 
pressure  design.  To  continue  design  for  overflow  conditions, 
use  formulae  of  Stage  II,  (6),  et  seq.,  of  Series  F,  and  to 
continue  ice-pressure  design,  use  formulae  of  Stage  II, 
et  seq.,  of  Series  E,  of  Chapter  III. 

After  a  section  has  been  finally  determined,  it  may 
have  fitted  to  it  circular  curves  of  suitable  radii,  or  tan- 
gents, to  simplify  construction. 

A  radius  equal  to  the  maximum  head,  with  center  on 
the  crest  line,  often  proves  suitable  for  first  trial  near 
the  crest,  down-stream.  Further  adjustment  of  the  upper 
part  of  the  parabolic  face  curve,  just  down-stream  of  the 
crest,  is  also  often  advantageous.  Here,  a  little  cutting 
away  of  the  curve  by  appropriate  curves  of  longer  radii, 
where  the  general  direction  of  the  lower  nappe  of  the  water 
sheet  is  nearer  the  horizontal  than  lower  down,  will  tend 
to  increase  the  flow.  There  is  less  chance  of  serious  results 
from  slight  vacuum  formation  at  this  point  than  further 
down,  and,  besides,  the  tendency  to  vibration  from  such 
a  cause  would  be  in  a  vertical  direction,  more  or  less,  with 
no  serious  consequence  to  the  structure.  Care,  however, 
should  be  taken  that  the  adjusted  curves  should  be  smoothly 
continuous  and  follow  the  sweep  of  the  parabolic  face, 
entering  the  sheet  without  abrupt  change  at  any  point, 
so  that  the  falling  sheet  of  water  would  take  its  further 
course  without  violence. 

For  a  low  dam,  the  parabolic  curve  down  the  face 
may  be  replaced  by  a  tangent  to  the  curve  at  a  point 
where  the  water  is  well  on  its  more  vertical  course. 


HIGH    MASONRY   DAM    DESIGN  119 

The  probable  patlj  of  the-^heet  of  water  should  always 
be  studied  in  connection  with  any  design. 

If  the  lower  portion  of  the  face  is  stepped,  the  study 
concerns  the  flow  upon  and  from  each  step.  The  para- 
bolic face  or  ogee  dam  is  of  advantage  in  this  respect, 
as  a  preliminary  study  of  the  stepped  cross-section,  since 
it  shows  the  minimum  sized  shape  for  stability,  upon 
which  the  steps  may  be  arranged  to  suit  the  flow.  In  this 
connection,  compare  Figs.  19  and  20. 

Sufficient  masonry  should  be  placed  just  down-stream 
of  the  toe,  depending,  to  great  extent,  upon  local  con- 
ditions. The  least  thickness  of  this  masonry,  other  con- 
siderations being  equal,  may  be  approximated  by  ascertain- 
ing the  probable  depth  of  back-water  just  below  the  dam, 
after  the  velocity  is  reduced,  beyond  the  shallower  dis- 
charging sheet.*  The  difference  in  head  between  the  upper 
surface  of  the  sheet  and  the  level  of  the  water  of  greater 
depth  further  down-stream  could  cause  an  uplift  if  the 
head  becomes  active  beneath  the  toe  protection.  The 
thickness  of  the  protection  masonry  should  be  at  least 
sufficient  to  balance  this  head  by  the  weight  of  this 
masonry. 

The  force  of  the  flow  from  the  toe  is  sometimes  broken 
up  by  masonry  baffles;  or  the  discharge  into  the  still 
back-water  below  the  dam,  the  presence  of  which  is  pro- 
vided for  the  purpose,  may  be  so  directed  by  the  curve 
of  the  dam  face  that  the  same  object  is  attained. 

*  In  this  connection  see  paper  on  "  The  Hydraulic  Jump  in  Open  Chan- 
nel Flow  at  High  Velocity,"  by  Karl  R.  Kennison,  in  Proc.  Am.  Soc.  of 
C.  E.  for  Sept.,  1915- 


120  HIGH  MASONRY  DAM  DESIGN 

SHAPE  OF  THE  FALLING  SHEET  OF  WATER. 
Spillway  Dam. 

A  study  of  the  water  sheet,  besides  outlining  a  proced- 
ure, will  indicate  its  possibilities  and  limitations — limita- 
tions to  the  procedure  because  the  available  data  are  not 
so  satisfactory  or  complete  as  might  be  wished  with 
respect  to  their  application  to  the  purpose  in  hand. 

Most  investigators  have  concentrated  almost  wholly 
upon  determination  of  the  discharge,  the  question  of  the 
shape  of  the  water-sheet  receiving  comparatively  little 
attention,  except  incidentally  to  the  effect  of  its  fluctua- 
tions upon  the  value  of  the  discharge.  An  attempt  at 
an  approximation,  only,  can  be  made  in  this  study,  which 
will  be  based  largely  upon  the  results  of  M.  Bazin's  re- 
searches.* 

M.  Boussinesq,f  in  an  account  of  his  investigations, 
in  1887  emphasized  the  importance  of  the  relation  of  the 
shape  of  the  under  side  of  the  sheet  to  the  contraction 
of  the  sheet  at  the  crest,  he  making  it  the  basis  of  a  new 
theory  of  flow  over  weirs. 

So  M.  Bazin,  with  great  care,  determined  the  profiles 
both  of  the  upper  surface  and  the  lower  surface  of  the 
sheet  for  two  sharp-crested  weirs,  one  3.7  feet  high  and 
the  other  1.15  feet  high  for  heads  varying  from  6  to  nearly 
1 8  inches. 

By  reducing  the  co-ordinates  of  the  curves  of  the  upper 

*  Bazin,  H.  "Experiences  nouvelles  sur  1'^coulement  en  deVersoir,  Annales 
des  Fonts  et  Chausse"es,  M6moires  et  Documents,"  1888,  1890.  See  also 
translation  by  Arthur  Marichal  and  J.  C.  Trautwine,  Jr.,  Proc.  Engineers' 
Club  of  Philadelphia,  Vol.  IX,  No.  3  and  Vol.  X,  No.  2. 

t  "  Comptes  rendus  de  1' Academic  des  Sciences,"  July  4,  1887. 


HIGH    MASONRY -*  DAM    DESIGN 


121 


0.1  0.^  0.3   0.4   0.5  0.6  0.7   0.8   0.9  1.0    1.1   1.2  1.3   1.4 

Nappes  for  Vertical  Weir   

"        «   Inclined  Weir _. 

Pig.  22 

FIG.  22    and    FIG.  23. 


122 


HIGH    MASONRY   DAM    DESIGN 


and  lower  surfaces  to  a  common  scale,  expressing  them  as 
ratios  of  the  head,  in  each  case,  he  established  that  for 
each  value  of  an  abscissa  there  is  a  corresponding  and 
sensibly  constant  value  of  the  ordinate. 

The  axes  of  rectangular  co-ordinates  pass  through  the 
crest,  Figs.  22  and  23,  at  A,  horizontal  abscissas,  x,  positive 

TABLE  III 
VALUES  FOR  SHAPES  OF  NAPPES 


Ordinates'  £r 

0 

A  V* 

X 

b' 

Crest  Vertical. 

Crest  Inclined  Down-stream.* 

Upper  Nappe. 

Lower  Nappe. 

Upper  Nappe. 

Lower  Nappe. 

-3.00 

0.997 

— 

_ 

—  I.OO 

0.963 

— 

— 

o.oo 

0.851 

o.ooo 

(0.730) 

O.OOO 

0.05 

0.059 

O.IO 

0.826 

0.085 

(0.700) 

(o.oii) 

0.15 

0.  101 

O.2O 

0-795 

o.  109 

(0.666) 

(0.005) 

0.25 

(0.778) 

O.II2 

0.30 

0.762 

0.  Ill 

(0.630) 

(-0.014) 

Q..35 

0.106 

0.40 

0.724 

0.097 

(0.585) 

(-0.044) 

0-45 

0.085 

0.50 

0.680 

0.071 

(0-535) 

(-0.083) 

0-55 

0.054 

0.60 

0.627 

0.035 

(o  .  480) 

(-0.130) 

0.65 

0.013 

0.70 

0.569 

—  0.009 

(0.418) 

0.80 

(0.507) 

(-0.068) 

(0.350) 

0.90 

(0-437) 

(-0.129) 

(0.276) 

I  .00 

(0.360) 

(0.196) 

I.IO 

(0.276) 

(0.109) 

1  .20 

(o.i  86) 

(0.009) 

1.30 

(0.085) 

(-0.098) 

Numbers  in  parentheses  have  been  scaled  from  Bazin's  plotted  profiles. 

*  Slope  i  on  2,  down-stream.     The  discharge  is  increased  by  nearly  13%  over  vertical 
weir's  discharge. 


HIGH   MASONRY   DAM   DESIGN  123 

jk 

to  the  right,  or  down-stream,  and  negative  up-stream; 
the  ordinates,  y,  positive  upward  from  the  crest  level  and 
negative  downward. 

These  co-ordinates  refer  only  to  the  curves  of  the 
nappe,  and  not  to  the  face  curve  of  the  dam. 

Ordinates  for  the  shapes  of  the  nappes  as  tabulated, 
likewise  are  vertical  and  positive  upward;  abscissae,  hori- 
zontal and  positive  to  the  right,  both  referred  to  the 
sharp  crest  of  the  weir,  or  "  theoretical  crest  "  of  Fig.  21, 
as  origin  of  rectangular  co-ordinates. 

The  ratios  in  Table  III,  if  multiplied  by  the  head  &', 
in  feet,  will  give  the  corresponding  locations  of  points  on 
the  respective  nappes,  in  feet,  for  a  given  case. 

The  table  gives  the  values,  extended  from  tabulated 
values  of  Bazin  from  his  profiles,  and  were  used  in  plotting 
Fig.  22. 

The  values  for  the  higher  weir  were  found  to  be  very 
precise,  by  comparison  of  18  different  determinations.  The 
final  values  are  here  given  and  are  to  be  multiplied  by  the 
head  bf  for  any  given  case. 

The  values  for  a  weir  whose  crest  is  inclined  downstream 
on  a  slope  of  i  on  2  are  also  included. 

In  addition  to  the  shape  determinations,  the  velocity 
and  pressure  heads  in  the  sheet,  at  the  contracted  section 
of  the  sheet,  were  carefully  found  experimentally.*  The 
velocities  are  plotted  in  Fig.  24  and  curves  were  drawn 
through  them.  Their  sensible  parallelism  is  significant,  in- 
dicating simple  proportionality  for  different  heads.  These 


*  M.  Bazin,  "Annales  des  Fonts  et  Chauss£es,  M£moires  et  Documents," 
1890.. 


124  HIGH  MASONRY  DAM  DESIGN 

1,02  Upper  Nappe 


BAZIN'S  EXPERIMENTS 
VELOCITIES  OF  FILAMENTS 

IN 

CONTRACTED  SECTION   OF  SHEET 
(sharp  crested  weir) 


wer  Nappe 


0.093 
0.074 
0.055 


0.0 


4  5  G  7 

Feet  per  Second 

FIG.  24. 


HIGH    MASONRY    DAM    DESIGN  125 

> 

velocities  were  determined  for  a  sharp-crested,  vertical 
weir,  3.7  feet  high,  with. six  different  heads,  ranging  from 
5.9  to  15.75  inches.  As  in  the  preceding  experiments, 
the  observed  factors  were  found  to  vary  proportionately 
with  the  head. 

The  purpose  of  this  study  is  primarily  to  establish, 
for  the  general  formulae  of  design,  preceding,  a  value  for 
the  parameter  of  the  parabolic  down-stream  face  of  the 
weir,  encroachment  within  the  water-sheet  to  be  assured. 
For  this  purpose  the  water-sheet's  lower'  surface  must 
be  traced,  that  is,  the  probable  path  of  the  sheet  must 
be  extended.  The  term  "  nappe  "  has  been  applied  to 
the  falling  sheet;  its  application  to  the  upper  and  lower 
surfaces  of  the  sheet  is  also  permissible.  These  may  be 
called,  then,  the  "  upper  "  and  "  lower  "  nappes,  for  con- 
venience. 

The  cases  of  vertical  and  inclined  weirs  depicted  in 
Fig.  22  indicate  the  effect  upon  the  upper  and  lower  nappes 
of  inclining  a  vertical,  sharp  crest  down-stream.  The  weir 
shown  in  Figs.  19  and  20  should  preferably  conform  to  the 
lower  nappe  of  the  inclined  weir,  rather  than  to  that  of 
the  vertical  weir.  On  the  other  hand,  high  overfall  dams 
usually  have  a  vertical  up-stream  face,  so,  for  this  type, 
the  nappe  of  the  vertical  weir  of  Fig.  22  should  be  em- 
ployed to  shape  the  crest.  As  the  formulas  of  design  are 
general,  with  regard  to  the  distance  of  the  actual  crest 
down-stream  of  the  vertical  up-stream  face,  and  to  the  par- 
abola of  the  down-stream  face,  any  condition  may  be  cared 
for  by  such  proper  considerations  of  the  flow  sheet  in  con- 
nection with  determining  the  parabolic  parameter  for  a 
given  case. 


126  HIGH    MASONRY   DAM   DESIGN 

As  an  aid  to  judgment,  it  may  be  stated  that  Bazin 
showed,  from  his  experiments  on  sharp-crested  weirs,  both 
vertical  and  at  various  inclinations  up-  and  down-stream  : 

I  e'  \ 

(i)  That  the  thickness   [7  of  Fig.  21  1   of  the  nappe 


over  the  crest  diminishes  as  the  weir  is  inclined  down- 
stream. This  diminution,  barely  perceptible  for  weirs  in- 
clined up-stream,  becomes  much  greater  when  we  pass 
to  weirs  inclined  down-stream. 

(2)  For  a  given  inclination  of  weir,  ^  increases  with 

b 

the  head,  &',  or  rather  with  the  ratio  of  b'  to  the  height  of 
the  weir. 

(3)  Except   for   very   low   weirs,    or   weirs   where   the 
velocity  of  approach  is  more  perceptible,  the  ratio  of  the 

•if        r 

height  of  the  lower  nappe  to  b'  or  the  value  of  —r-f  —  (see 
Fig.  21)  appears  to  be  independent  of  the  head  bf  for  a  given 

7    /  1 

inclination  of  weir.     M.   Boussinesq  also  stated        ,      to 

be  a  constant. 

(4)  The    thickness    of    the    sheet    measured    over    the 
summit  of  the  lower  curve  (at  the  crest  line  in  Fig.  21) 
increases  from  the  greatest  observed  inclination  up-stream 
to  that  of  i  on  i,  or  45°,  down-stream,  beyond  which  the 
thickness  diminishes. 

(5)  For  a  weir  of  constant  height,  the  ratio  of  the  height 
of  the  lower  nappe  (bf  —  b,  of  Fig.  21)  to  b'  diminishes  as 
the  inclination  of  the  weir  changes  from  up-stream  to  down- 
stream.    The  ratio  of  its  chord  length  to  b'  (the  chord 
being  measured  horizontally  from  the  "  theoretical  "  crest 

of  Fig.  21)  decreases  as     ,  .     decreases. 

b 


HIGH  -MASdNRY   DAM   DESIGN  127 

'  Jfc 

(6)  For  weirs   inclined   up-stream,  the  inclination,  al- 

7    /  T 

though  it  modifies  considerably  the  value  of  ,   has 

b 

e> 
but  little  effect  upon  that  of  —n  which  does  not  vary  as 

much  as  0.02  as  we  pass  from  a  vertical  weir  to  an  up- 
stream inclination  of  45°.  For  weirs  inclined  down-stream, 

e> 

however,  —.  diminishes  rapidly,  and  the  curve  of  the  sur- 
o 

face  becomes  elongated  as  the  inclination  increases. 

Further  experiments   by   Bazin   on   weirs   of  irregular 

section*  indicated  that,  for  different  heads, f  the  ratio  —f 

did  not  change  very  rapidly  for  the  same  weir,  and  that 
the  ratio  of  the  thickness,  e ',  of  the  sheet  of  water  at  the 
up-stream  edge  of  the  sill,  to  the  total  head,  &',  varies 
within  very  extended  limits  for  weirs  with  sloping  faces. 
This  variation  is  due  to  three  causes,  as  follows : 

(a)  The  "width"  of  the  crest,  or  ratio  of  the  head, 

bf,  to  the  crest  width.     For  squared  timbers,  for  example, 
e' 

the  value  —,,  quite  large  for  small  heads,  diminishes  as  the 
b 

head  increases,  approaching  progressively  to  that  which 
applies  to  the  nappe  of  a  sharp-crested  weir  ("in  thin 
partition  ").  This  influence  ought  naturally  to  be  found 
again  in  weirs  with  sloping  faces,  the  crest  having  a  fixed 
width. 

(b)  Inclination  of  the  down-stream  face.     When  slightly 
inclined  to  the  horizontal  it  exercises  an  influence  similar 

*"Annales   des   Fonts  et   Chausse"es,   Me"moires  et   Documents,"   1898, 
2me  trimestre,  pp.  121-264. 

t  Heads  from  o.i  to  0.4  meter  (3.9  to  15.7  inches). 


128  HIGH   MASONRY   DAM    DESIGN 

to  enlarging  the  crest  and  has  the  same  effect  in  increas- 

e> 

ing  the  value  of  -=-7. 
b 

(c)  Inclination  of  the  up-stream  face.     The  effect  of 

this  inclination,  which  modifies  the  contraction  of  the  sheet 

e' 
of  water  at  the  passage  of  the  sill,  is  to  dimmish  -.-7. 

These  results  for  inclination  of  the  faces  *  are  analogous 

to  those  established  for  sharp-crested  weirs;    but  it  was 

e' 
observed  that  the  values  of  —n  instead  of  growing  with  the 

head,  as  for  sharp-crested  weirs,  continue  to  decrease  as 
for  weirs  of  squared  timber,  thus  fixing  the  influence  of 
width  of  crest.  From  another  point  of  view,  it  may  be 

observed  here  that  the  effect  of  this  virtual  widening  of 

0t 

the  crest  tends  to  increase  the  initial  values  of  —„  over  the 

6 

corresponding  values  for  a  weir  of ~  sharp  crest,  and  these 

e> 
values  of  ^7  with  increase  of  head,  bf,  tend  to  approach  the 

e' 
values  of  —r  for  a  sharp-crested  weir,  hence  a  comparative 

diminishing  of  values  of  jj  as  the  head  is  increased,   may 

result. 

From  a  further  series  of  experiments,  one  set  of  which 
was  upon  a  weir  with  a  crest  15.7  inches  wide  and  with  up- 
stream face  at  a  constant  slope  of  2  on  i,  and  a  down- 
stream face  inclination  varying  from  i  on  2,  to  i  on  4  and 

T  on   6,  it  was  developed  that  the  value  of  —r  remained 
*  Crests  of  weirs  varied  from  nearly  4  inches  to  nearly  8  inches  in  width. 


HIGH   MASONRY  DAM   DESIGN 
* 

nearly  unchanged  and  its  value  ranged  between  0.87  and 
0.88. 

Weirs  with  faces  joined  by  arcs  of  circles  to  a  hori- 
zontal crest,  and  above  all  weirs  with  curved  profile,  con- 

e* 
duce  to  small  values  of  7-7,  but  it  should  be  remembered, 

in  this  connection,  that  the  thickness  ef,  measured  properly 
at  the  "  theoretical  crest  "  or  where  the  sill  is  entirely 
replaced  by  an  edge,  is  not  at  all  comparable  with  that 
observed  at  the  most  elevated  point  of  the  lower  nappe. 

On  weirs  where  the  crest  is  joined  to  the  up-stream  face 

e' 
by  a  curved  surface  —,  (presumably  measured  at  the  actual 

crest)  reduces  to  below  0.80;    but  the  crest  is  not  then  a 

thin  edge. 

e' 

Free   nappes    in    "thin  partition"  give  for  -77  values 

b 

ranging  from  0.85  to  0.86.  The  foregoing,  considered  in 
connection  with  Fig.  22,  will,  it  is  believed,  indicate  the 
effect  upon  the  shape  of  the  nappes  of  the  inclination 
of  the  up-stream  face,  the  breadth  of  crest,  and  the  down- 
stream face  slope  near  the  crest. 

Although  objection  may  be  raised  against  employing 
empirical  data  from  observations  of  comparatively  low 
heads  for  conditions  that  might  obtain  for  high  heads, 
it  should  be  remarked  that,  as  the  head  increased,  both  for 
the  sharp-crested  as  well  as  for  the  irregularly  shaped  weirs, 
the  variations  of  the  relations  approached  better  definition 
than  for  the  low  heads.  Also,  there  should  be  noted  the 
constant  ratios  that  obtained  in  some  of  the  phases  out- 
lined. In  the  absence  of  more  extended  data,  therefore, 
the  several  indications  suggested  above  may  be  employed. 


130  HIGH   MASONRY  DAM   DESIGN 

1 

It  will  suffice  to  consider,  for  example,  the  sharp- 
crested  vertical  weir,  in  connection  with  Figs.  22,  23,  and  24. 
The  problem  remains  to  extend  the  nappes,  as  plotted 
by  Bazin's  co-ordinates,  in  terms  of  the  head  bf.  In  Fig. 
23  verticals  through  the  tenth  marks,  from  i'  to  7',  have 
been  drawn. 

The  filament  curve,  OPE,  of  the  average  velocity  is 
then  traced,  its  position  in  the  vertical  (the  crest  line) 
through  the  actual  crest  being  first  ascertained. 

An  integration  of  each  of  the  velocity  curves  of  Fig.  24, 
between  the  upper  and  lower  nappes  and  the  vertical  axis 
of  co-ordinates  to  which  it  is  referred,  yields  the  discharge 
for  each  case  (within  about  5  per  cent  of  the  values  obtained 
by  the  Francis  weir  formula,  Q  =3-33L//3/2).  Each  dis- 
charge divided  by  the  vertical  distances  between  the  nappes 
at  the  section  (Fig.  24)  will  give  the  value  of  the  average 
velocity,  whence  its  ordinate,  or  location  above  the  actual 
crest,  may  be  read  from  its  velocity  curve.  This  location 
was  found  to  vary  from  41  to  44  per  cent  of  the  water- 
sheet's  thickness,  above  the  actual  crest.  The  average 
for  the  six  heads  observed  was  found  to  be  43  per  cent.  * 

The  value  of  the  coefficient  C  in  the  formula  for  dis- 
charge, Q,  given  in  Part  I  of  this  chapter,  lies  between 
3.40  and  3.50.  The  average  was  found  to  be  3.48  in  the 
cases  given.  H  is  taken  upon  the  "  theoretical  "  crest. 

As  has  been  brought  out,  the  ratio,  — —  (Fig.  21), 
is  practically  constant.  The.  filaments  passing  the  section 

*  Mr.  Richard  Muller,  "  Engineering  Record,"  Oct.  24,  1908,  uses  \\  but 
this  value  appears  to  involve  pressure  heads  rather  than  velocities  in  their 
distribution  through  the  sheet. 


HIGH  MASQNRY   DAM   DESIGN  131 

Jk 

X—X',  Fig.  23,  are  "sensibly  parallel.     Bazin  demonstrated 

that  the  slight  inclination  in  their  direction  did  not  affect 
the  readings  of  the  instruments  obtaining  the  measure- 
ments of  the  velocities.  Hence,  we  may  proceed  to  lay 
off  on  each  of  the  verticals  i  —  i',  2  —2',  etc.,  0.43  (i  —  i') 
0.43  (2—2'),  etc.,  above  the  curve  of  the  lower  nappe. 
The  sheet  section  lines,  of  which  a  — a'  is  one,  are  next 
drawn  by  making  each  pass  through  a  point  on  a  vertical, 
such  as  P,  just  located,  and  normal  to  the  straight  line 
joining  the  two  similar  points  adjacent  to  P.  On  these 
lines,  a  — a,  0.43  of  the  distance  of  each,  comprehended 
between  the  upper  and  lower  nappes,  is  laid  off  above 
the  lower  nappe.  These  may  be  considered  points  on  the 
path  of  the  filament  OPE  of  average  velocity. 

The  discharge  divided  by  the  thickness  of  the  sheet, 
such  as  at  a  — a',  will  give  the  average  velocity  which 
may  be  plotted  to  scale  as  Vp  in  Fig.  23,  normal  to  a  — a', 
and  applied  to  the  point  of  the  filament  curve  nearest 
P.  The  vertical  and  horizontal  components  plotted,  will, 
from  their  resulting  values,  enable  the  origin,  0,  of  the 
filament  curve  to  be  located  with  sufficient  precision.  This 
may  be  tried  by  consideration  of  other  points  and  an 
approximate  location  fixed.  It  will  be  found  that  the 
average  velocity  of  the  section  X—Xf  applies  here,  though 
the  horizontal  component  V*  should  be  used  for  the  sheet 
extension  beyond  section  a  — a'.  With  0  and  Vh  fixed, 
the  path  OPE  may  be  approximately  computed  with 
reference  to  0  and  plotted,  by  choosing  points  down  the 
curve  and  computing  the  vertical  component  of  the  velocity 
attained,  by  the  laws  of  falling  bodies.  Laying  off  this 
vertical  component  for  each  point,  and  plotting  Vh,  results 


132  HIGH  MASONRY  DAM    DESIGN 

in  obtaining  the  velocities  of  the  sheet,  tangential  to  OPB, 
for  the  various  points.  The.  discharge  divided  by  the 
tangential  velocity  at  each  point  gives  the  thickness  of 
the  sheet.  This,  laid  off  on  a  line  normal  to  the  tan- 
gential velocity  and  through  the  given  point,  with  0.43 
below  the  point,  and  0.57  of  the  thickness  of  the  sheet 
above  the  point,  determines  two  points,  the  one  on  the 
lower  nappe,  and  the  other  on  the  upper  nappe.  This 
procedure  may  be  continued  as  far  down  as  desirable  and 
smooth  curves  drawn  through  the  points,  so  determined, 
will  give  the  direction  and  shape  of  the  sheet  with  suf- 
ficient accuracy. 

A  parabola  whose  parameter  is  about  1.86',  or  K  =  i.8, 
will  approximate  the  position  of  the  lower  nappe.  Its 
origin  is  at  the  highest  point  of  the  lower  nappe,  in  the 
crest  line,  therefore,  for  a  cross-section  to  be  well  within 
the  sheet,  its  K  should  be  greater  than  K  =  i.S.  K  =  2.2$ 
will  provide  a  parabola  that  will  meet  this  requirement 
adequately  and  allow  the  curve  of  the  face  between  the 
crest  and  7'  (Fig.  23)  to  be  somewhat  flattened  by  a  curve 
of  longer  radius,  as  suggested  earlier. 

A  parabolic  curve  for  the  nappes  of  the  inclined  weir 
shown  in  Fig.  22  could  be  similarly  worked  out. 

It  is  reasonable  to  suppose  that  the  friction  of  an 
actual  masonry  surface  upon  which  the  lower  nappe  is 
flowing  would  modify  the  thickness  of  the  sheet.  This 
would  be  probable,  from  a  consideration  of  the  similarity 
in  effect  of  the  crest  of  masonry  Jhere  and  the  broad  crest 
effects  on  the  thickness,  cited  earlier  from  experiments; 
however,  the  curves  of  the  face  would  tend  to  have  an 
opposite  effect  from  that  caused  by  the  virtual  widening 


HIGH   MASONRY   DAM    DESIGN  133 

of  the  crest;  but  to  "what  extent  cannot  be  said  at  present. 
It  is  possible  that  the  horizontal  component  of  the  velocity 
near  the  crest  would  be  cut  down  in  value  from  that  ob- 
taining with  discharge  free,  and  over  a  sharp  crest;  but, 
in  basing  the  determination  of  a  cross-section  upon  the 
flow  over  a  sharp  crest,  there  is  obtained  a  larger  trajectory 
for  the  falling  sheet  than  might  otherwise  result,  and 
so  a  preferable  cross-section  will  result,  as  it  is  on  the  safe 
side,  both  as  to  encroachment  within  the  water  sheet  and 
stability. 


CHAPTER  VII 
THE  ARCH  DAM 

BEFORE  deciding  upon  the  cross-section  for  a  masonry 
dam,  the  proposed  site  should  be  carefully  studied  with 
regard  to  its  topography,  to  determine  the  type  of  struc- 
ture that  can  be  most  advantageously  used  with  existing 
local  conditions,  keeping  in  mind  especially  the  question 
of  economy  of  material. 

It  is  evident  that  under  all  circumstances  the  choice 
must  lie  between  the  gravity  type  of  dam  heretofore  dis- 
cussed and  any  one  of  the  arched  types  about  to  be  briefly 
touched  upon.  But  where  gorges  or  canyons  are  en- 
countered, the  selection  of  the  arch  most  naturally  suggests 
itself,  especially  since  in  those  of  200  to  500  feet  in  width 
moderate  spans  result. 

The  economy  of  one  form  over  another  will  depend 
upon  whether,  with  the  greater  length  but  smaller  cross- 
sectional  area,  the  arched  type  will  require  more  or  less 
material  than  the  straight  gravity  type  with  its  shorter 
length  but  greater  cross-sectional  area. 

Should  the  gravity  section  be  discarded  in  favor  of 
the  other  form,  the  further  question  arises  as  to  which 
of  the  various  arched  types  may  be  used  to  best  advantage, 
and  it  is  the  purpose  of  the  few  following  paragraphs  to 
refer  briefly  to  these  types  before  proceeding  to  the  dis- 
cussion of  the  design  of  what  may  be  termed  a  simple 
arched  dam. 

134 


HIGH    MASONRY   DAM   DESIGN  135 

Buttress  Arch  Type.  —  This  form  of  dam  consists  of 
a  series  of  plain  or  reinforced  concrete  arches,  either  ver- 
tical or  inclined  down-stream,  supported  at  the  abutments 
by  buttresses,  and  may  need  special  attention,  particularly 
if  the  length  of  the  proposed  structure  is  considerable, 
to  determine  whether  it  may  be  economically  used  in  a 
given  location.  Such  an  investigation  may  be  made  by 
first  considering  the  arches  in  connection  with  the  loading 
producing  the  stresses  in  them,  and  second  by  analyzing 
the  buttresses  with  respect  to  the  component  arch  thrusts 
transmitted  to  them  and  acting  in  a  down-stream  direction. 

The  former  investigation  would  be  undertaken  by  means 
of  any  of  the  prevailing  arch  theories  applicable  to  the 
case  in  hand,  while  the  latter  would  be  prosecuted  by  the 
use  of  the  formulae  already  established  in  connection  with 
the  gravity  type  of  dam.  To  apply  these  formulae  it  is 
necessary  to  consider  a  unit  of  thickness  of  the  buttress 
and  to  reduce  the  component  down-stream  thrust  of  two 
adjacent  arches  acting  upon  a  single  buttress  to  an  equiv- 
alent hydrostatic  pressure,  so  that  the  conditions  may 
correspond  to  those  in  the  design  of  the  gravity  type  dam, 
where  the  water  pressure  is  taken  as  acting  over  a  unit 
length  of  the  dam. 

This  reduction  may  be  accomplished  by  calculating  an 
equivalent  value  of  f,  the  weight  of  a  cubic  foot  of 
water,  whereupon  with  the  thickness  of  the  buttress 
assumed  for  each  level  or  stage,  and  with  the  masonry 
density  either  assumed  or  known,  the  formulae  for  design 
may  be  employed  directly  to  solve  for  the  successive  lengths 
of  base,  /,  of  the  buttress.  The  sides  of  the  buttresses  at 
each  stage  may  be  assumed,  for  ease,  to  be  vertical  planes, 


136  HIGH    MASONRY   DAM    DESIGN 

until  the  calculations  are  completed,  when  a  proper  batter 
may  be  given  to  avoid  the  successive  offsets  which  would 
result.  It  may  be  advisable  also  to  investigate  the  but- 
tresses as  "  plates  "  to  determine  their  probable  tendency 
to  "  buckle."  One  of  the  chief  advantages  of  this  type 
of  dam,  as  is  quite  evident,  is  its  comparative  freedom  from 
the  effect  of  uplift. 

Gravity  Section. — A  gravity  section  may  be  arched  in 
plan,  or  a  special  cross-section  for  the  arch  type  may  be 
developed,  since  the  arch  action  may,  except  for  high 
stresses,  be  limited  in  the  thicker  cross-section.  In  the 
gravity  sections  arched  in  plan,  it  may  be  shown  that  the 
arch  takes  from  5  to  8  per  cent  of  the  load,  or  expressed 
differently,  that  the  cantilever  transmits  most  of  the  load 
to  the  foundation. 

Arch  Section. — The  section  of  an  arch  dam  shown  in 
Fig.  25  was  investigated  by  the  method  described  later, 
and  the  study,  including  other  dams  of  like  dimension, 
except  for  down-stream  face  batters  and  bases,  led  to  the 
conclusion  that  by  thickening  the  top  to  at  least  5  or  6 
feet,  thus  gradually  increasing  the  thickness  until  it  reached 
the  section  of  Fig.  25  at  a  depth  of  about  80  feet,  a  section 
could  be  obtained  which  would  reduce  the  arch  stresses  at 
the  top  without  increasing  them  below  at  all  appreciably. 

The  above-mentioned  figure  represents  in  a  general 
way  the  cross-section  of  an  arched  dam,  and  by  the 
curves  indicates  the  amount  of  arch  action  at  the  various 
levels.  It  is  evident  from  the  curves  that  at  the  founda- 
tion no  arch  action  exists.  The  up-stream  radius  was 
taken  as  350  feet,  and  the  arch  span  as  600  feet. 

To  obtain  a  tentative  cross-section  for  an  arch  dam, 


HIGH    MASONRY    DAM    DESIGN 


137 


the  thin  ring  formula  may  be  used  to  calculate  the  bases 
at  the  different  levels. 


I  =  length  of  base  in  feet  ; 
P  =  water  pressure  in  pounds  per  square  foot  ; 
Rn  =  length  of  the  up-stream  radius  in  feet  ; 

q  =  average  stress  in  the  ring  in  pounds  per  square  foot. 

Compression  -  _ 


140       \ 


Cross-section 


0        10 
Tons  per  Square  Foot 


FIG.  25. 


138  HIGH    MASONRY   DAM    DESIGN 

The  Constant  Angle  Dam. — It  may  be  shown  that  to 
obtain  an  arch  dam  of  minimum  volume,  when  the  struc- 
ture acts  as  an  arch,  and  with  minimum  stresses  (even 
near  the  foundation),  any  arch  slice  must  be  subtended 
by  a  horizontal  central  angle,  between  abutments,  of 
I33°~34/-*  Practically,  this  angle  maybe  reduced  to  120°. 
The  fact  that  this  type  of  dam  has  an  ability  to  act  as 
an  arch,  to  a  much  greater  degree  than  the  ordinary 
arch  dam,  follows  from  the  fact  that  an  arch,  when 
loaded,  undergoes  a  deflection  proportional  to  the  square 
of  the  up-stream  radius  (see  Eq.  8,  page  149),  and  in  the 
"  constant  angle  "  type,  this  radius  may  be  several  times 
shorter  at  the  foundation  than  at  the  top.  As  a  con- 
sequence, the  deflection  at  the  base  required  for  the  same 
unit  stress  would  be  proportionately  less  than  the  deflec- 
tion required  at  the  top  to  produce  this  same  unit  stress, 
and  as  canyons  are  generally  narrower  at  the  bottom  than 
at  the  top,  this  condition  usually  applies. 

The  principle  underlying  the  "  constant  angle  "  may 
be  developed  as  follows : 

From  the  formula  just  given : 

j_PRn 

~ 

it  is  evident  that  the  base  /  and  therefore  the  cross-sectional 
area,  varies  directly  as  the  radius  Rn.  The  volume  in  a 
given  section,  however,  is  equal  to  the  area  times  the  length 
of  the  mean  arc,  which  latter  may  be  expressed  in  terms 
of  the  length  of  the  mean  radius  (designated  as  Rm) 

*  "  The  Constant  Angle  Arch  Dam,"  by  Mr.  Lars  R.  Jorgensen.  Trans. 
Am.  Soc.  C.E.,  Vol.  78,  p.  685. 


HIGH    MASONRY    DAM    DESIGN  139 

times   the   subtended   angle  ^designated   as    2<f>n,  expressed 
in  terms  of  TT)  .     We  may  therefore  write  for  the  volume  V  : 


But  Rm,  the  mean  radius,  is  equal  to  one-half  the  span, 
or  width  of  canyon  W,  divided  by  sin  #„,  or,  expressed  as 
an  equation* 


- 

sin  <f>n 

Since  the  area  of  the  cross-section  is  proportional  to 
the  length  of  the  radius,  Rn,  and  the  volume  to  Rm,  we 
may  write  without  sensible  error  : 


sin2  <f>n         sin2  </>„ 

This  expression,  in  which  C  and  K  are  constants,  K 
depending  upon  the  width  of  the  canyon,  indicates  that  the 

volume  varies  with  the  term  -7—? — • 

sin2  <f>n 

If  we  differentiate  this  expression  with  respect  to  $„, 
and  equate  the  differential  coefficient  to  zero  for  a  mini- 
mum, we  have, 

i 

2  COt  (j>n 

which  equation  is  satisfied  by  the  value  of  66°- 47'  for  <£„, 
whence  2<£w  equals  i33°-34/. 

Poisson's  ratio  for  lateral  strains  is  taken  into  con- 
sideration in  determining  the  relative  arch  action  in  a 
constant  angle  type,  a  value  of  one-fifth  being  adopted, 
and  the  initial  stresses  induced  axially  by  the  weight  of 


140 


HIGH    MASONRY    DAM    DESIGN 


the  dam  on  the  foundation,  together  with  the  water  load, 

being  utilized  to  help  support  the  latter. 

Fig.  26  represents  a  cross-section  of  a  dam,  developed 

by   the    constant-angle    principle,    250   feet   high,    with   a 

base  of  70  feet  and  a  cross-sec- 
tional area  of  9,668  square  feet. 
An  equivalent  rectangle  of  equal 
base  would  have  to  be  138  feet 
high. 

If  the  specific  gravity  of  the 
concrete  be  assumed  at  2.3,  the 
average  vertical  pressure  may  be 

expressed  as  —  —  ,  where  H  is  the 
a 

height  of  the  dam,  and  a  is  the 
ratio  of  the  total  height  to  that 
of  the  equivalent  rectangle,  whence 

81. 


138 


The  mean  vertical  compression 
for    this    section    would    then  be 

2  iH 

—  —  —  =i.27H   with    the    reservoir 
1.81 

empty,  expressed  in  terms  of  the  head  of  water,  when  the 
latter  is  at  an  elevation  of  the  top  of  the  dam. 

With  the  reservoir  full,  the  radial  water  pressure  is 
assumed  to  counteract  the  strain  of  the  masonry  acting 
in  an  up-stream  and  down-stream  direction,  although  there 
is,  of  course,  no  direct  opposing  force  on  the  down-stream 
side,  acting  horizontally  up-stream. 

It  is  reasonable  to  assume,  however,  that  the  reactions 


1  700  000  Lb. 

FIG.  26. 


HIGH    MASONRY   DAM    DESIGN  141 

at  the  abutments  at  least  'partially  provide  an  indirect 
force  acting  from  the  down-stream  side  in  an  up-stream 
direction. 

As  Poisson's  ratio  is  approximate,  the  full  head  is  used 
as  active  in  this  respect,  hence  the  total  resulting  initial 
axial  compression  at  the  foundation  for  the  section  may 
be  written  as 


The  height  of  water,  represented  by  h,  that  this  initial 
axial  compression  of  o.^$H  will  resist  without  causing 
the  arch  at  the  bottom  to  shorten  in  length,  will  result 
by  use  of  the  ring  formula 

j_PRn 

*  ~  -  > 

q 

in  which  q  is  represented  by  0.4  5  #  and  P  by  h,  whence 


For  the  narrow  section  of  Fig.  26, 


hence  this  section,  by  application  of  the  above  expression, 
is  able  to  carry  as  an  arch 

/*=o.45#X  —  =  0.42^, 

/  0 

or  42  per  cent  of  the  total  head  of  water,  before  any 
shortening  in  the  length  of  the  arch  occurs.  The  re- 
maining 58  per  cent  of  loading  is  distributed  between  the 
arch  and  cantilever  as  explained  later  on. 


14:2  HIGH    MASONRY    DAM    DESIGN 

Arched  Dam  Investigation. — After  the  plan  and  cross- 
section  of  an  arched  dam  have  been  settled  upon,  the 
structure  may  be  investigated  to  determine  the  proportion 
of  the  loading  resulting  from  horizontal  water  pressure 
which  will  be  cared  for  by  the  structure  acting  as  a  hori- 
zontal arch,  and  that  cared  for  by  the  structure  acting 
as  a  vertical  cantilever,  respectively.  From  this  resulting 
distribution  of  loading,  the  value  of  the  intensities  of  stress 
on  vertical  planes  normal  to  the  axis  of  any  arch  ring  under 
consideration  may  be  calculated  from  the  resulting  thrusts 
into  the  sides  of  the  canyon.  Stress  intensities  for  hori- 
zontal joints  may  be  found  by  combining  the  stresses 
due  to  the  horizontal  loads  assumed  by  the  vertical  canti- 
lever, acting  within  the  elastic  limit  of  the  material,  with 
the  stresses  due  to  the  weight  of  the  dam  above  the  joint 
in  question. 

This  investigation  for  distribution  of  loading  between 
arch  and  cantilever  may  be  made  independently  of  the 
value  of  the  modulus  of  elasticity  of  the  concrete  or  material 
of  which  the  dam  is  constructed,  if  considered  homogeneous, 
as  will  appear. 

Limitations. — If,  however,  the  dam  be  built  in  sections 
with  transverse,  vertical  "  contraction  "  joints,  dividing 
it  segmentally  into  portions  approaching  voussoirs  in  their 
nature,  it  could,  under  certain  conditions,  hardly  be  con- 
sidered to  act  as  an  elastic  arch.  These  joints  may  be 
open  more  or  less  at  times,  according  to  the  atmospheric 
temperature,  the  season  of  the  year  when  masonry  between 
them  was  laid,  and  the  depth  of  water  behind  the  dam, 
with  its  consequent  effects  of  swelling  the  masonry  and 
affecting  its  internal  temperature.  Furthermore,  the  con- 


HIGH    MASONRY   DAM    DESIGN  143 

traction  at  any  time*  may  be  greater  at  the  top  than  further 
down,  or  within  the  dam  and  different  segments  may  be 
simultaneously  in  different  conditions  of  stress  due  to 
contraction. 

In  short,  the  conception  of  the  dam  as  a  horizontal 
arch,  fixed  at  the  ends  and  along  the  foundation  of  the 
dam  and  acting  to  a  greater  or  less  degree  as  a  huge  vous- 
soir  arch,  involves  both  a  consideration  of  internal  tem- 
perature conditions  and  a  knowledge  of  the  value  of  the 
modulus  of  elasticity  of  the  great  mass  of  the  dam,  both 
of  which  are  sufficiently  uncertain  to  render  anything  but 
an  extended  investigation  under  various  assumptions  of 
doubtful  value. 

But  the  limits  between  which  the  behavior  of  the 
structure  may  lie,  viz.,  that  of  an  elastic  arch,  held  at  the 
sides  and  bottom  of  the  gorge,  and  that  of  a  cantilever, 
bearing  the  total  load,  may  therefore  be  profitably  in- 
vestigated. 

Again,  the  deflection  of  such  cantilever  with  an  assumed 
value  for  the  modulus  of  elasticity  of  the  material  may 
be  considered,  together  with  the  deflection  of  the  top- 
most arch  slice  at  the  crown,  due  to  the  opening  of  the 
contraction  joints.  These  may  be  compared,  or  they  may 
be  reduced  to  a  corresponding  temperature  range  and 
compared  with  the  maximum  possible  range  at  the  site 
of  the  dam,  whence  an  indication  as  to  the  probability 
of  arch  action  ceasing  wholly  or  in  part  may  be  reached. 

Method  of  Arch  and  Cantilever  Analysis. — For  the  con- 
sideration of  the  elastic  arch  with  cantilever  action  and 
no  contraction  joints,  or  joints  tightly  closed,  the  fol- 
lowing method  is  elaborated  from  a  method  outlined  by 


144 


HIGH    MASONRY    DAM    DESIGN 


the  late  R.  Shirreffs  in  a  discussion  of  a  paper  on  the  Lake 
Cheesman   Dam   and   Reservoir   by   the   late   Charles   L. 


Origin  for  Joints  1,2,3,4 


FT 


QT 


t     if-  ^U 


-*T-Br<fc-H 


*     Pr 


Joint  ( n 


— ir=u7aT-r- 
n=_7)__     __J 


Computed 
position 

J 


EXAMPLE: 

(i)  For  battered  up-stream  face, 


-d2)  tan  03  +  (d4-</2)  tan  8t  +  (d6-da)  tan  05 

-d$  tan  Ot  +  tfi—dt)  tan 


(2)   For  vertical  up-stream  face  (assume  Joint  5  to  be  base  of  dam,  6&  =  0°),  then, 

D2  =  A2+A3+A4-|-(d3-d2)  tan  Bi+(dt-dd  tan  04. 
Note:  All  joints  may  be  referred  to  any  one  origin. 


FIG.  27. 


HIGH    MASONRY  DAM    DESIGN  145 

Harrison  and  Mr.  -Silas  P^Woodard,  Members  Am.  Soc. 
C.E.* 

The  formulae  developed  in  the  following  pages  are  ap- 
plicable to  the  vertical  cantilever,  contained  between  two 
vertical,  radial  planes  (i  foot  apart  at  the  extrados),  of 
any  arched  dam,  either  of  overfall  (spillway)  or  complete 
retaining  type. 

The  down-stream  face,  though  it  may  be  curved  in 
vertical  profile,  should  be  considered  straight  between 
load-points.  Thus,  in  a  spillway  dam,  the  load-points 
at  the  more  curved  portions  of  the  vertical  poofile  may 
be  taken  nearer  together.  The  load-points  may  be  ar- 
bitrarily chosen  in  positions  just  so  that  the  portions 
between  successive  load-points  may  be  considered  as  trap- 
ezoids  without  essentially  altering  the  cross-section  of  the 
dam.  The  "  joints  "  are  taken  at  the  load-points.  Gen- 
erality has  been  attained  by  introducing  the  expression  for 
the  moment  of  inertia  of  the  horizontal  cross-section  of 
this  vertical  cantilever  in  terms  of  the  variable,  x,  before 
integrating. 

The  following  nomenclature,  together  with  that  shown 
in  Fig.  27,  applies. 

NOMENCLATURE 

For  the  Cantilever. — The  " origin"  of  a  joint  is  the  point 
of  intersection  of  the  down-stream  side,  or  face,  of  the 
dam,  next  below  that  joint,  with  the  corresponding  up- 
stream face  of  the  dam,  both  produced,  if  necessary. 

B  =  the  batter  of  the  down-stream  side  (or,  if  the  up- 
stream side  is  battered  also,  the  combined  batter 

*  Trans.  Am.  Soc.  C.  E.,  Vol.  53,  p.  155. 


146  HIGH    MASONRY   DAM    DESIGN 

of  the  down-stream  and  corresponding  up-stream 
side)  next  below  the  joint,  or  load-point,  in 
question. 

00=  the  vertical  distance   from  the  origin  of  any  joint 

of   the   cantilever  to   any  level  in   the  portion 

immediately  below  the  joint  (between  it  and  the 

next  joint). 

dn  =  the  vertical  distance  of  the  joint  n  below  its  origin 

(d  signifies  general  expression). 

l=Bx=the  length  of  a  horizontal  joint  of  masonry  at  the 
depth  x  below  the  origin  for  that  joint   (same 
as  thickness  of  arch  ring). 
/B=the  length  of  a  horizontal  joint  of  masonry  at  the 

depth  dn. 
ZnJM=the  length  of  a  horizontal  joint  of  masonry  at  the 

depth  d»+i,  etc. 

m  =the  total  number  of  load-points,  or  joints. 
n=the  number  of  the  load-point,  or  joint,  considered, 

beginning  with  the  top  load-point. 

Q»=the  total  load  of  water,  in  tons,  considered  as  con- 
centrated  at   any   joint,  n,  over  i    foot   length 
of  extrados.     (See  left-hand  diagram,  Fig.  27.) 
Pn  =  that   part  of  Qn  assumed  by  cantilever  action  at 

the  center  of  the  dam. 

E  =  modulus  of  elasticity  of  the  material  of  the  dam. 
I  =  the  moment  of  inertia  of  the  horizontal  cross-section 
of  the  cantilever,  at  the  level,  x.     (See  Appendix 
I  for  derivation.) 


n          -  6RnB4x 

'       '     '     ' 


HIGH    MASONRY   DAM   DESIGN  147 

where  Rn  is  the  radius  of  the  arch  extrados  at  the 
portion  of  the  cantilever  to  which  %  is  taken. 
Mn  =  the  moment  of  all  loads  above  the  joint  considered 
(joint  n),  (M  signifies  such  moment  in  general). 

An  =  the  deflection  of  each  individual  portion  of  the 
cantilever  produced  by  all  of  the  loads  above 
that  portion  (see  Fig.  27),  (A  signifies  such  de- 
flection, in  general). 

On  =  the  angle  of  deflection  (change  of  angle)  at  any 
load-point,  n,  having  reference  only  to  the 
portion  of  the  cantilever  between  that  load- 
point  and  the  one  next  below  (n  +  i).  (See  Fig. 
27).  (0  signifies  such  angle  in  general.) 
A,=the  total  deflection  of  the  cantilever  at  any  load 
point,  n. 

For  example,  from  Fig.  27, 

Dz  =  A2  -f-  A3  -|-  A4  -j-  ( ds  —  dz)  tan  63  -f-  (d±  —  d%)  tan  0± 

for  a  dam  whose  up-stream  face  is  vertical.     (See  p.  144.) 
gn  =  the  vertical   extent    of   hydrostatic  pressure,  the 

resultant  of  which 

(Q»)  is  concentrated  at  the  joint  considered,  or  load- 
point  n. 

k'n  =  the  portion  of  g»  above  joint  n 
kn  =  the  portion  of  g»  below  joint 
(7'»=the  head  of  water  on  the  level  at  the  upper  end 

of  the  distance  gn. 

£n=the  head  of  water  on  the  level  at  the  lower  end 
of  the  distance  gn.     (For  example,  G\  =G'z.) 

The  following  two  expressions    (readily   derived)    give 
the  relations  among  gn,  k»,  k'n,  G'n,  and  Gn\ 


j-or  gn  =&„+&'„.    (2) 


148  HIGH    MASONRY    DAM    DESIGN 


'n  +  [f  (G'n  -k'n)}2  -I  (G'n  -fe'.),       .;    <•  .    (3) 

f(G-n+fen),       .     .     (4) 


also, 

,/  .  <?'„ 


/r/ 
3(6- 


and  are  convenient  for  computing  the  hydrostatic  con- 
centrations for  load-points  wherever  chosen.  (See  Fig.  27.) 

Eq.  (40)  may  be  used  to  locate  the  last  concentratipn 
at  the  lower  part  of  the  dam. 

Qn,  in  tons,  may  be  computed  from  Eq.  (5),  following: 


if  G'n  and  gn  are  expressed  in  feet. 

For  the  Arch.  —  (Arch  ring  depth  taken  at  i   foot.) 
.Rn=the  radius  of  extrados  at  load-point  n.     (If  radius 
of    extrados    is    constant,    throughout,    Rn=R.) 
In  equations,  the  extrados  radius  will  be  desig- 
nated by  Rn. 
rn  =the  radius  of  the  axis  of  the  arch  at  any  load-point,  n. 

rn=Rn--  .........  (6) 

2 

gn=unit   loading   on   axis   of   arch    (corresponding   to 
extrados  unit  loading)  at  load-point  n. 

jvo^v  ...  ...  (7) 


=i  central  angle  of  arch  span  at  level  of  load-point,  n. 
=  thickness  of  arch  ring  (depth  i  foot)  at  any  level 
of  dam, 


HIGH    MASONRY    DAM    DESIGN 


149 


ln  =same  at  level  dn  =  length  of  joint  n. 
Dc  =  deflection  at  crown  of  arch  at  level  of  load-point 
(cantilever  "  joint.")  n.     (See  Appendix  II.) 

2  sin  4>n 


Dc  = 


-(l  -COS   <£„)  +  COS2   (j>n  - 


1.40 


1.35 


1.30 


1.25 


1.20 


1.15 


g  1.10 


1.05 
1.00 
0.95 
0.90 
0.85 
0.80 


3<t>n 

sin  fa 


-cos  fa) 


+  COS  fa -4 


2  sin  <j>n 


(1  —  cos  <t>n)  +cos2  4>n  - 1 


CC, 


-fl-cos  4>t, 


+cos  <f>n  -  4 


40° 


50  ( 


70° 


qnrn 
El, 


,  (8) 


FIG.  28. 

The  trigonometric  function  of  fa,  in  Eq.  (8),  may  be 
denoted  by  CCC  and  can  be  plotted  as  a  curve,  greatly 
simplifying  its  application,  in  the  calculations.  (See  CCC 
of  Fig.  28.) 


150  HIGH    MASONRY   DAM    DESIGN 

The    Cantilever.  —  The   general   differential   expressions 
for  flexure  of  a  cantilever  may  be  written 


_ 

........     do) 


in  which  there  may  be  substituted 

Mn=P1(x-dl)+P2(x-d2)+ 
which  may  be  transformed  to 


.  .  .   +Pndn).     (n) 

Also,  there  may  be  written,  for  an  arch  dam  with  ver- 
tical up-stream  face 


-dn)  tan  6n+2-\-(dn+B-dn)  tan  0n+3  +  .  .  . 
+  (dm-dn)tanem  ............     (12) 

Where  the  up-stream  face  is  battered,  as  shown 
below  joint  No.  5  in  Fig.  27,  let  m'  be  the  number  of 
the  joint  at  the  base  of  the  vertical  portion,  or  joint  5, 
as  there  shown.  Eq.  (12),  with  this  special  significance 
applied  in  this  case  to  m',  will  hold  as  written,  down  as 
far  as  the  base  of  the  portion  with  vertical  up-stream 
face,  if  n5  be  changed  to  m'. 

The  additional  terms,  made  up  of  differences  of  d's 
times  respective  tangents  of  0's,  will  be  successively  of 
the  form, 

tan  Bm,+2+  .  .  . 

-dm_l)  tan  Bm\ 


HIGH    MASONRY   DAM   DESIGN  151 

* 

or,  for  a  dam  with  fart  vertical  and  part  battered  up-stream 
face,  as  shown  in  Fig.  27, 


+  (dn+2-dn)  tan  6n+2  +  (dn+3-dn)  tan  0n+3  +  .  .  . 
+  (dm,-dn)  tan  Om,  +  (dm,+l-dm.)  tan  8m,+l 

tan  0W'+2  +  .  .  .  +(dm-dm_1)  tan  0ro.  (120) 


In  case  there  is  no  vertical  up-stream  face  (see  lower 
portion  of  right-hand  diagram,  Fig.  27)  Eq.  (i2a)  would 
take  the  following  form  : 


-dn+i)  tann+2  +  .  .  .    +(dm-dm_l)  tan  Om.     (126) 

Tan  0,  in  Eqs.  (12),  (i2a)  and  (126)  above,  may  be  con- 
sidered equal  to  0,  in  each  case. 

Cantilever  Deflection  Equations.  —  By  substituting  in 
Eqs.  (9)  and  (10),  above,  the  expressions  for  /  and  M, 
given  in  Eqs.  (i)  and  (n),  respectively,  and  integrating 
with  respect  to  x,  the  resulting  expressions  between  the 
limits  x=dn+i  and  x=dn,  and  reducing,  there  may  be 
obtained  the  respective  expressions  for  An  and  0n  (Eqs. 
(13)  and  (14)).  (See  Appendix  I.) 

By  means  of  Eqs.  (13)  and  (14),  p.  153,  the  A  and 
6  for  each  joint  or  load-point  can  be  computed,  and  com- 
binations, as  indicated  by  any  one  of  Eqs.  (12),  (120), 
or  (126)  that  pertains,  made  for  each  joint,  whence  there 
results  for  each  such  joint  an  equation  involving  E,  and 
the  various  P's  (the  latter  being  the  unknowns)  for  the 
deflection,  Dn. 


152  HIGH    MASONRY   DAM    DESIGN 

Arch  Deflection  Equations. — Eq.  (8),  just  preceding, 
provides  for  writing  the  several  expressions  for  the  crown 
deflections  of  the  arch  elements  of  the  dam,  one  expression 
for  each  of  the  load-points  assumed.  These,  too,  will 
be  in  terms  of  E  and  the  various  P's,  as  an  inspection  of 
Eq.  (7)  in  connection  with  Eq.  (8)  will  show. 

Resulting  Simultaneous  Equations.  —  By  equating  each 
expression  for  the  arch  crown  deflection,  to  the  expression 
for  the  cantilever  deflection  at  the  same  level,  or  load  con- 
centration-point, a  series  of  simultaneous  equations  may 
be  evolved,  E  dividing  out  each  time,  and  PI  to  Pm 
appearing  in  each  of  the  m  equations  as  the  unknown 
quantities. 

Solution  for  Distribution  of  Loading  between  Arch  and 
Cantilever  Actions. — A  solution  of  this  set  of  simultaneous 
equations  will  yield  the  value  of  the  P  for  each  joint  or 
load-point.  These  loads  assumed  by  the  cantilever  action 
may  then  be  subtracted  from  the  respective  total  hydro- 
static loadings  (Q)  whence  the  amount  assumed  by  each 
of  the  horizontal  arch  laminae  results. 

The  fact  is  neglected,  however,  that  the  several  arch 
slices  throughout  the  dam  actually  cannot  move  freely, 
in  relation  to  each  other.  This  would  tend  to  stiffen 
the  dam  along  the  arch  axis  and  thereby  transmit  the 
arch  thrusts  in  an  axial  direction  from  a  given  arch  into 
the  abutment  of  some  lower  arch  slice. 

The  general  expressions  for  Ekn  and  Edn  follow  as 
Eqs.  (13)  and  (14).  These,  with  equations  for  I  and 
DC1  (Eqs.  (i)  and  (8)),  are  derived,  as  outlined  above,  in 
Appendices  I  and  II. 


HIGH    MASONRY   DAM    DESIGN  153 


Rnlndn+i  Wn+l 

ln  +l  —  f->  -L 

log- 


A.)  ( 


n+  i 


6(d2n+l  ~dn2)        2  - 


X2. 30259  log 
l°g-T^ 


.      -.     .     (x4) 


154  HIGH    MASONRY    DAM    DESIGN 

These  expressions  involve  only  P,  d,  B,  R,  I,  and  E. 

In  use  of  expressions  (13)  and  (14)  above,  for  any 
joint,  n,  to  which  a  particular  batter,  B,  naturally  applies 
(with  reference  to  that  joint's  particular  origin),  care  should 
be  taken  that  the  various  d's  (cf.  P\d\,  P^d^  above)  are 
referred  to  that  particular  origin  of  the  given  joint  n. 

It  will  usually  be  found  that,  in  Eq.  (14)  (for  E0»), 
two  terms,  the  fourth  in  the  first  pair  and  the  third  in  the 
last  pair  of  braces  are  negligible. 

Application  of  Foregoing  Formulae  (13),  (14)  and  (12), 
(i2a),  or  (i2b),  (cantilever  equations) .-- The  procedure 
will  be  conveniently  described  by  referring  to  forms,  in 
order,  for  tabulating  the  factors  that  enter  into  the  con- 
struction of  the  cantilever  equations  of  deflection;  that 
is,  the  application  of  the  foregoing  formulae  (13),  (14), 
and  (12),  (i2a),  or  (126).  Numerical  values  are  given 
for  illustration  only,  and  to  facilitate  comprehension  of  the 
use  of  the  tabulation  forms. 

The  first  step  after  the  cross-section  has  been  fixed, 
is  to  choose  the  load -points,  or  joints.  In  an  overfall 
dam  these  will  be  more  numerous  at  the  upper  and  lower 
portions,  where  the  curvature  of  the  down-stream  face 
necessitates  shorter  tangents  to  approximate  more  nearly 
the  curved  cross-section  by  one  of  rectilinear  sides,  forming 
a  series  of  trapezoids,  with  the  joints.  The  load-points 
may  number  from  4  to  10,  according  to  the  type  of  dam 
and  its  height  and  shape. 

Second,  calculate  the  g's  by  means  of  Eqs.  (3)  or  (4) 
and  for  the  last  joint,  m,  locate  the  concentration  for  the 
distance  gm,  remaining,  by  means  of  Eq.  (40).  (See  Fig.  27.) 

Third,   from  the  foregoing,   compute  by  Eq.    (5)   the 


HIGH    MASONRY   DAM    DESIGN 


155 


Q's.  The  heads  on  the  load-points  may  be  entered  into 
the  tabulation  as  well,  together  with  the  lengths  of  the 
joints  (Ts). 

These  may  be  tabulated  conveniently  as  in  Table  V. 


TABLE  V 


»- 

G'n 

*. 

, 

G,4° 

Sn 

32 

Qn 

(tons) 

Head  on 
Qn 

In 

I 

* 

2 

* 

3 

* 

m 

* 

*  Computed. 

The  fourth  step  is  to  compute  the  origins  of  the  joints 
(see  nomenclature  for  definition  of  origin).  These,  once 
found,  should  be  placed  as  shown  in  Table  VI,  entitled 
"  Distances  of  Origins  above  Joints,"  together  with  these 
distances,  which  are  useful  in  carrying  out  the  provisions 
of  origin  reference,  noted  immediately  after  and  referring 
to  use  of  Eqs.  (13)  and  (14). 

Suppose  the  position  is  desired  of,  say,  Joint  No.  2 
with  reference  to  the  origin  of  Joint  No.  3.  In  the  left- 
hand  column  of  Table  VI,  No.  3  gives  the  line  and  2  the 
column,  and  in  line  3  and  column  2  is  read  44.63.  That  is, 
the  origin  of  Joint  No.  3  is  44.63  feet  above  joint  No.  2, 
or  No.  2  is  44.63  feet  below  the  origin  of  Joint  No.  3. 
Where  these  distances  are  negative,  the  distance  d  in  the 
formulae  must  be  written  with  the  minus  sign. 


156 


HIGH    MASONRY    DAM    DESIGN 


TABLE   VI 
DISTANCES  OF  ORIGINS  ABOVE  JOINTS  (IN  FEET). 


Joint  No. 

i 

2 

3 

8 

9 

Base. 

I 

12.18 

21.73 

9-55 

2 

20.26 

29.81 

37.62 

17.36 

1      3 

•a 

35-08 

44-63 

52.45 

64.26 

29.18 

5    L 

| 

8 

-41.42 

-31.87 

-24.05 

65.34 

75-34 

116.76 

9 

-79.98 

-70.43 

-62.61 

26.78 

36.78 

21.76 

V 

TABLE   VII 

COEFFICIENTS  FOR  0,  IN  EQUATIONS  FOR  D 
(Table  of  Differences) 


di 

dz 

d3 

d* 

ds 

dg 

<*, 

9-55 

d* 

o   .  .... 

17-37 

7.82 

4J 

W             ^ 

«.l 

<N 
J|- 

29.19 

19.64 

11.82 

• 

"^      : 

^H 

+ 

^              i 

rfs 

106.76 

97.21 

89-39 

77-57 

^9 

116.76 

IO7.2I 

99  39 

87-57 

... 

IO.OO 

Base 

121.76 

112.  21 

104.39 

92.57 

15.00 

5-0 

HIGH    MASONRY   DAM    DESIGN 


157 


At  this  step  in  the  work,  the  coefficients  for  8,  in  canti- 
lever equations  for  D,  might  be  compiled,  as  indicated 
in  Table  VII. 

For  instance,  in  computing  D2,  it  is  necessary  to  use 
as  a  factor,  among  others,  d^—d^.  From  Table  VII,  as 
given  there,  for  example,  the  value  19.64  feet  corresponds. 

What  is  really  the  fifth  step  is  to  compute  the  factors 
occurring  in  Eqs.  (13)  and  (14)  as  tabulated  below,  in 
Table  VIII. 

Each  large  factor  is  here  designated  by  a  Roman 
numeral,  for  future  reference. 

TABLE   VIII 


Joint 
No. 

(D 

(II) 

(Ill) 

(IV) 

2.30.259,        dn  +  l 

6(<Wi  ~dn) 

"(<Wi-<y 

Wn+i-<V> 

K>            g  '  /v 

Rn                 dn 

lnRndn+l 

dn+l 

V2»+i 

I 

0.0013898 

0.0027862 

etc. 

etc. 

2 

O.OOI553I 

0.0018624 

5.032262 

1-0303393 

3 

etc. 

etc. 

etc. 

etc. 

etc. 

Joint 
No. 

(V) 

(VI) 

B 

B. 

B. 

2.30259  , 

Rn 

2.30259, 

VJ>Kn 

Rn 

D           loa 
Rn 

/n-(3  +  V^w 

/n-(3-v3-)K» 

I 

etc. 

etc. 

etc. 

etc. 

etc. 

2 

—  0.00000481 

—0.0000181 

0.09285 

o  .  00862 

0.000800 

3 

etc. 

etc. 

etc. 

etc. 

etc. 

etc. 

158  HIGH    MASONRY    DAM    DESIGN 

Also  tabulate  for  reference,  the  values  of  the  following : 

Rn,  Rn2, 

/— 

2  ~ 


Tabulate  for  the  various  joints  -,  —  ,  and  —  ,  the  first 

-D      ^)  JD 

two  of  these  three  quantities  for  calculating  the  0's,  and 
the  last  two  for  the  A's.     These  could  be  added  in  three 

extra  columns  to  Table  VIII.   -^  and  —  could  be  multiplied 

JD  nz 

by  the  various  d's  and  tabulated,  to  facilitate  calculations 
of  EA's  below. 

Eqs.  (13)  and  (14)  may  now  be  written  in  terms  of  (I), 
(II),  (III),  etc.,  of  Table  VIII,  with  substitutions  for  /„ 
and  Rn\  that  is,  if  the  portions  within  the  braces  of  Eqs. 
(13)  and  (14),  involving  (I),  (II),  (III),  (IV),  etc.,  be 
designated  by  Sn  and  S'n  and  S"n  and  5//rn,  there  may 
be  written  : 


EA3  = 

-D3  -D3 

etc.  etc. 

The  SVs  and  5n's  for  all  the  joints  should  first  be 
separately  computed  by  aid  of  Table  VIII,  and  then 
combined  in  the  last  written  expressions,  above. 


HIGH    MASONRY   DAM    DESIGN  159 

Similarly, 


3  +-^(Pi  +  P2  +P3)S"2, 
.D3  ±>3 

etc.  etc. 

The  S'"n's  and  S"n's,  should  be  separately  computed 
as  for  S'n  and  5n,  in  advance. 

The  resulting  expressions  for  EAi,  EA2,  etc.,  with 
£0i,  E02,  etc.,  will  each  now  consist  of  n  terms;  for  example, 
EA3  or  £63  will  each  have  3  terms,  involving  Pi,  P2,  and 
P3  with  numerical  coefficients. 

Example  : 

The  computation  for  E&2,  for  instance,  may  be  arranged 
as  follows  : 

There  will  be  nine  terms  involving  (I),  (II),  (III), 
(IV),  (V),  and  (VI),  of  Table  VIII.  Assume  /2=3.86, 
B2  =0.09285,  ^2=350- 

Substituting  in  Eq.  (13), 


-(7oo-/n)(II)+(IV)+443-8o-°-26V) 

O  J 


x[(42oo-6/B)  (I)  -(III)  -(2100-1.  268/»)(V) 
-(2ioo-4.722/n)(VI)]  .....     .-'.....     (15) 


160 


HIGH    MASONRY   DAM    DESIGN 


S'n  and  Sn  comprise  the  quantities  within  the  two  sets 
of  brackets  and  their  determination  is  illustrated  in  Table 
IX.  For  E&2,  £'2  and  £2  are  here  shown,  for  example,  the 
quantities  of  Joint  2,  in  Table  VIII,  being  used  for  (I), 
(II),  etc. 


TABLE   IX 
COMPUTATION  OF  5'2  AND 


For  EA2 


2100-15.44 

X(i)2 

350 

—0.0092561 

-(200-3.86)X(II)2 

—  1.2964911 

+  (IV)2 

+  1.0303393 

For  S't 

,  443-  80-0.  268X3-  86  ^ 

1                 350                (X)2 

16562-3.732X3-86 

X(V  L)z      — 
350 

5', 

-0.2754930 

+  (4200-6X3-  86)  X(I)2 

+6.4870502 

-(IH)2 

-5.032262 

-(2ioo-i.268X3.86)(V)2 

+0.0101005 

For  S2 

-(2100-4.732X3-86)  (VI)2      = 

+0.0377100 

52 

+  1.5025987 

assuming,  for  sake  of  explanation,   values   for  di, 
25.5384  and  41.5384,  respectively,  or 


of 


HIGH    MASONRY   DAM    DESIGN  161 

'  > 

115. 994478^1X25. 5384 +P2X4i-5384)( -0.2754930) 

+  1249. 2i924(Pi+P2)(  +  i. 5025987) 
£A2= -816.0966^1 -1327.3873^2  +  1877. o752(Pi+P2) 

=  io6o.9786Pi  +549.6879P2. 
E&2  should  be  similarly  calculated. 

A  separate  sheet  should  be  devoted  to  the  computation 
of  each  EAn  and  each  E6n,  so  that  ready  reference  may  be 
made  thereto. 

The  A's  and  0's,  having  been  thus  computed,  should 
next  be  collected  according  to  formula  (12),  or  (120)  or 
(126),  as  applicable,  and  by  aid  of  Table  VII. 

To  this  end,  the  following  form  of  tabulation  (Table  X) 
for  each  Dn  will  prove  convenient,  for  obtaining  and  sum- 
ming coefficients  of  the  unknowns,  P,  assuming,  for 
example,  five  load-points. 

Application  of  Eq.  (8).  (Arch  Equation.) — Having  writ- 
ten the  expressions  for  cantilever  deflections  (ED 5,  ED 4, 
.  .  .  EDi)  as  indicated  in  Table  X,  it  remains  to  con- 
struct the  equations  for  the  deflections  of  the  corre- 
sponding arches,  by  the  application  of  Eq.  (8). 

The  tabulation  of  Table  XI  will  expedite  the  formation 
of  the  arch  deflection  equations,  shown  in  the  last  col- 
umn of  Table  XL 

Eq.  (8)  is  based  upon  the  assumption  of  ends  fixed  for 
the  arch.  Limitations  to  which  this  assumption  may  be 
subject  have  been  pointed  out  previously  in  relation  to 
arch  action  as  a  whole;  but  the  usual  excavations  for  the 
dam  into  the  rock  sides  of  the  gorge  or  canyon,  justify  to 
some  extent,  this  assumption. 

For  the  determination  of  the  angles  4>n  of  the  various 


162 


HIGH    MASONRY    DAM    DESIGN 


So 

Tf 

IO  vO     t^  00 

ON 

^2: 

J^> 

oo_ 

00     10 
^t"  rx 

ro 

00                                    t^ 

to 

^ 

to  oo 

o 

vo"                      o" 
»o 

to 

D!? 

ON 

t^ 

¥> 

ON                              ON 

00 

vd 

VO      Ol 

1 

:      l 

i 

O     O^ 

ON    C? 

HH        -rj- 

ON  vO 
(S     CO 

ON 

§         s  g, 

vO 
| 

ON   <NJ 

N  2 

CO    d 

! 

0) 

rO                        t^-   O 

•-I                                   CO    Tt- 

I 

$£ 

II 

10 

£ 

ro                00    VO   OO 
O                   Tj-   O     HH 

Q\                      vO     VO      Tt" 
00                      M     IN.    rj- 

1 
IO 

ro 

en 

» 

ON    9 

ON    rt- 

ro 

6 

CO                          VO       >O     HH 

Q 

«s 

HH      ,£• 

00     CN 

HH       CN 

g. 

"cu 

l^»                   ^t1   rO    n 
>O                  «O   H     t^ 

fc- 

1*3 

+  + 

-4^ 

+ 

x   ;, 

9    I 

^_    ^ 

^ 

O             <N     t^.    <N     0 
N              CN     d     IO  00 

g 

TAB 

MPUTATIC 

* 

?  o 
o  3" 

t^   O 

^t-    ON 

vO    OO 

»O    ON 
Tj-   CO 

ON 

ON 

1 

CO           ON   iO    HH     ON 

ON           ON    ro  00     ON 
OO           00    00     »O    HH 

IO           O<    vO     ^    ON 

CO            HH    oo     IO  00 
VO             tO  VO     ON    ON 

g 

vO* 

d 

+  + 

+  + 

+ 

+           +  +  +  + 

+ 

-- 

t^*           00     O     tO    04 

M 

II 

11 

ON 

1 

o 

ON          ON    O     HH     cO 
O            ^^    ^O     O     ^O 
CO           ro    ro    to    O 
10           O     HH     ON    5 

5 
* 

O     £• 
OO     r^* 

10    ro 

I 

; 

N            10    ON    ro    ro 

OX                      HH        (VJ        (SJ        HH 

00 
ON 

\\   \\ 

II      II 

ii 

II         II    II    U    II 

4:1 

. 

Jj 

t3 

+  a 

3 

<S 

40  ^  „  „  ^, 

o 

ii 

<f  ^ 

Jt               ^    ^    *$    ^ 

§< 

Ji 

^ 

< 

+                   'III 

i 

HIGH    MASONRY    DAM    DESIGN 


163 


arch  laminae,  an  average  depth  of  excavation  (often  about 
15  feet)  may  be  assumed  on  the  profile  of  the  site.  This 
may  be  transferred,  by  means  of  excavation  contours,  to 
the  plan  of  site  and  points  thereon  determined  at  the  levels 
of  the  assumed  load-points,  or  joints.  These  points  are 
taken  on  the  foundations,  so  that  they  are  approximately 
midway  between  the  upstream  and  downstream  faces  of 
the  dam  at  their  respective  elevations  in  rock.  A  line 
on  the  plan  through  these  points  is  the  line  of  the  profile 
for  the  axes  of  the  various  arches,  whence  the  central 
angles  4>n  are  readily  obtainable.  They  are  to  be  entered 
in  the  second  column  of  Table  XI. 

The  trigonometric  function  of  </>n,   CCC,   may  be  read 
directly  from  Fig.  28,  p.  149. 

TABLE   XI 
ARCH  EQUATIONS 


Joint 
No. 

*n 

I 

2 

Rn 

«.-?-'. 

Rn 
rn 

Qn~Pn 

(tons) 

sn 

I 

59° 

I.I85 

350 

348-815 

1.003397 

0.56  -Pi 

6.0 

2 

etc. 

etc. 

etc. 

etc. 

etc. 

etc. 

etc. 

etc. 

Joint 
No. 

J 

«n  =  - 

*»  (Qn-rn\ 

Function 
of  $n 

=  ccc 

V 

ln 

EDC  (Eq.  (8)  ) 

rn  \       8n       ) 

Eq.  (7) 

I 

o.  09365  -0.16723.?! 

1.0952 

5I338.3562I 

5265.5657- 
94O2  .  7982^1 

2 

etc.    . 

etc. 

etc. 

etc. 

etc. 

164 


HIGH    MASONRY    DAM    DESIGN 


Simultaneous  Equations  Formed. — Equating  each  arch  de- 
flection to  its  corresponding  cantilever  deflection  (for  in- 
stance, ED i  of  Table  X  with  EDe  of  Table  XI,  for  Joint 
No.  i)  and  reducing,  eliminates  the  quantity  Ei  between  the 
two,  and  the  simultaneous  equations  (involving  P  with 
numerical  coefficients)  as  many  as  there  are  joints,  re- 
sulting, may  be  tabulated  as  indicated  in  Table  XII. 
These  simultaneous  equations  are  to  be  solved  for  the 
values  of  the  quantities  P,  in  tons. 

TABLE   XII 

FINAL  SIMULTANEOUS  EQUATIONS 
Coefficients  of  Quantities  P. 


Joint  No. 

PI 

Pz 

Ps 

I 

+  19249.45065 

+67709.40901 

+3077-35694 

2 

etc. 

etc. 

etc. 

3 

etc. 

etc. 

etc. 

4 

etc. 

etc. 

etc. 

5 

+80.40272 

+70.49011 

+     51-90397 

Joint  No. 

PI 

P6 

= 

I 

+905-599I6 

+80.85725 

+5265.56566 

2 

etc. 

etc. 

etc. 

3 

etc. 

etc. 

etc. 

4 

etc. 

etc. 

etc. 

5 

+29.60060 

+215.65413 

29155.10921 

For  nearly  all  of  the  above  computations,  a  calculating 
machine  will  be  found  most  expeditious. 

The  final  results  may  be  tabulated  in  some  such  form 
as  in  Table  XIII,  in  which  loads  are  in  tons  and  intensities 
of  stress,  in  tons  per  square  foot. 


HIGH    MASONRY   DAM    DESIGN 

il 

TABLE  XIII 
RESULTS 


165 


Joint 
No. 

Elev. 

Total 
Load 
Concen- 
tration 
«?„). 

Load  Assumed  by 

Percent. 
Arch 
Action. 

Maximum  Stresses  in 

Cantilever 

c*y. 

Arch 
(Qn-Pn>- 

Horizontal  Plane 
(Cantilever) 

Vertical 
Radial 
Planes 
(Arch)] 

Up- 
stream. 

Down- 
stream. 

I 

2 

3 

etc. 
Base 

96.0 
80.0 
SO.o 

14.0 

0.56 
12.55 
etc. 

etc. 

-1.87 
-4.11 
+4-14 

+  2.43 
+  16.66 

435 
133 

etc. 

+  13-9 
etc. 

—  ii.  7 
etc. 

etc. 
etc. 

Cantilever  Stresses. — To  get  the  resulting  maximum 
stress  intensities  in  any  horizontal  joint  of  the  cantilever, 
combine  the  average  intensity  of  stress  on  that  joint  due 
to  the  weight  of  the  superimposed  masonry  with  •  the 
stress  intensity  due  to  the  moment  of  that  masonry  plus 
the  moments  of  the  ^cantilever  loading,  P,  all  moments 
taken  about  the  center  of  gravity  of  the  horizontal  section 
of  the  cantilever  at  the  given  joint.  For  calculating  the  posi- 
tion of  the  centroid  of  the  horizontal  section  of  the  canti- 
lever at  any  joint  of  length  /„,  referred  to  the  down-stream 

edge  of  the   given  joint,  use  the    expression  —(3    n     n). 

3X2^-47 

Or,  this  resultant  moment  of  the  forces  about  the  centroid 
of  the  given  horizontal  section  of  the  cantilever  may  be 
thus  calculated  and  entered  into  the  well-known  expression, 

kl 
M=-j-,  as  written  on   page  29,  Eq.    (18),  or   using   here 

C  for  di  and  Sd  (down-stream  intensity)  and  Sv  (up-stream 
intensity)  for  k,  to  prevent  confusion,  the  last  expression 
may  be  written  in  the  form 

MC 


166  HIGH    MASONRY   DAM   DESIGN 

By  substituting  ln  for  Boo  in  Eq.   (i),  for  7,  reducing, 
and  inserting  in  Eq.  (140),  the  expression  for  S&  results: 


(  M 

nnn-nn 
and 


It  should  be  remembered  that  to  the  stress  intensities 
found  by  Eqs.  (146)  and  (14^)  should  be  added  the  average 
intensity  of  stress  due  to  the  superimposed  masonry,  to 
get  the  resultant  intensity,  as  previously  stated.  (In 
computing  the  weight  of  masonry,  it  will  be  sufficiently 
close  to  use  an  average  horizontal  base  in  each  case,  of 
rectangular  section  in  lieu  of  the  trapezoidal  bases.) 

Arch  Stresses.  —  To  get  the  resulting  maximum  stress 
intensity  on  the  vertical  radial  planes,  at  any  joint-level, 
n,  at  the  crown,  the  expression  for  Mc  should  be  employed, 
viz.  : 


12  \        0o        /  \30n+sm  0o  cos  0o  — 4 

and  the  maximum  stress  in  the  arch,  Sa,  is: 

6MC 


(14*) 


The  stress  resulting  from  Eq.  (14^),  for  any  level,  n, 
must  be  combined  with  the  axial  arch  thrust  (S/)  as  found 
byEq.  (i4/): 


HIGH    MASONRY    DAM    DESIGN  167 


If  the  stress  is  desired  al*  any  other  point  than  at  the 
crown,  the  expression  M  +,  in  Eq.  (12),  of  Appendix  II, 
should  be  employed. 

Arch  Dam  of  Rectangular  Cross-section.     Special  Case.  — 

The  foregoing,  it  will  be  remembered,  is  applicable  to 
a  dam  of  any  cross-section,  with  load  points  assumed  at  will. 

In  case  it  is  desirable  to  investigate  a  dam  of  rectang- 
ular vertical  cross-section,  the  load  points  being  chosen 
at  equal  intervals  a  vertical  distance,  a,  apart,  with  con- 
stant moment  of  inertia  for  the  horizontal  section  of  canti- 
lever, the  following  expressions,  Nos.  15  to  18,  apply. 

These  are  derived  in  a  similar  way  to  those  that  have 
preceded.  Some  slight  approximation  may  have  to  be 
made  in  computing  the  total  load  concentrations,  if  they 
be  considered  a  constant  distance  apart,  but  usually  a 
dam  of  rectangular  cross-section  is  not  very  high  and  this 
approximation  will  not  be  serious. 

Loads,  etc.,  are  located  with  reference  to  the  top  of  the 
dam  in  this  case,  with  dn  =na. 

-2a)  +  .    .    .    +PH(x-na).    .     (15) 


.   (16) 


+[i+a(*-i0JP.J.   (17) 
.    .    .     +Am)  +a[tan  0n+1 
+  2  tan  0«+2+3  tan  0n+3+  •    -    •   +(m—n)  tan  0ro].  (18) 
As  before  0  may  be  written  for  tan  0. 


168  HIGH    MASONRY   DAM   DESIGN 

Expressions  (16),  (17)  and  (18)  for  the  rectangular 
section  of  arched  dam  correspond  to  expressions  (13),  (14) 
and  (12),  respectively,  and  fulfill  the  same  offices.  Eq.  (8) 
still  applies,  for  the  arches  to  be  considered. 

The  derivation  of  Eqs.  (16)  and  (17)  are  compara- 
tively simple  and  are  outlined  in  Appendix  I. 

For  further  references  to  methods  of  analysis  of  arch 
dams,  the  reader  is  referred  to  Transactions  of  the  Ameri- 
can Society  of  Civil  Engineers,  Vol.  53. 


CHAPTER  VIII 

RECENT  CONSIDERATIONS  OF  THE  CONDITION  OF  STRESS 
IN  MASONRY   DAMS 

CONSIDERABLE  discussion  has  been  raised  within  the 
past  few  years,  by  criticisms  being  leveled  at  the  present 
general  procedure  in  the  design  of  high  masonry  dams. 
This  has  properly  perhaps,  been  more  pronounced  abroad 
than  in  this  country,  since  the  matter  may  be  said  to  have 
been  precipitated  by  the  publication  of  a  paper  by  Mr. 
L.  W.  Atcherley  of  London  University,  "  On  Some  Dis- 
regarded Points  in  the  Stability  of  Masonry  Dams."* 

It  is  the  purpose  to  outline  the  analysis  as  presented 
there,  and  to  call  attention  to  some  of  the  discussion  which 
followed,  in  order  to  indicate  the  status  of  the  theory 
involved  in  the  design  of  such  structures. 

The  paper  referred  to  takes  exception  to  current 
•practice  in  regard  to  the  matter  of  design  and  indicates 
a  need  for  both  re  vision .  and  extension  in  the  analysis, 
and  then,  supplementing  the  generally  accepted  ideas  as 
to  the  distribution  of  normal  stress  on  horizontal  planes, 
by  an  assumption  as  to  the  shear  on  these  planes,  proceeds 
to  show  that  peculiar  and  unexpected  conditions  arise. 


*  Dept.   of  Applied  Mathematics,   University  College,   University  of 
London.     Drapers'  Company  Research  Memoirs.     Technical  Series  II. 

169 


170  HIGH    MASONRY    DAM    DESIGN 

It  is  a  fact  that,  owing  ist,  to  the  manner  in  which 
masonry  structures  are  built,  i.e.,  of  a  mixture  of  stone 
and  cement,  and  2d,  to  the  nature  of  the  sections  at  the 
springings  or  areas  of  support,  it  is  practically  impossible 
to  apply  to  them  the  general  theory  of  elastic  bodies. 
Consequently,  the  treatment  as  it  is  employed  to-day 
has  been  developed,  but  only  by  the  use  of  certain  assump- 
tions which  it  may  be  shown  are  not  precisely  exact. 

The  basis  of  the  present  investigation  rests  upon  the 
four  following  formulae,  in  which  the  usual  distribution 
of  normal  unit  stress  on  horizontal  planes  is  accepted, 
but  to  which  is  added  an  assumed  condition  as  to  the 
distribution  of  horizontal  shear. 


«*-lT M 


Al'n 


T          =  *L[__T 

•*•  max         \  \    v          L  ]•>     • 


2A^       '  ' (4) 


c  =  distance  along  the  horizontal  joint  from  the  cen- 

troid  to  the  point  of  application  of  the  resultant. 

d  =  distance  along  the  horizontal  joint  from  the  cen- 

troid  to  the  point  locating  the  neutral  axis. 
£  =  length  of  the  horizontal  joint. 
Cmax  =  maximum  compressive  stress  on  the  joint, 
^max  =  maximum  tensile  stress  on  the  joint. 


HIGH    MASONRY   DAM   DESIGN  171 

iJp 

Q  =  vertical   component   of   the   resultant   force   acting 

on  the  joint. 
A  =  area  of  the  joint. 
5  =  shear  at  any  point  y  in  the  joint. 
P  =  total  shear  on  the  joint. 
y  =  the  distance  from   the  centroid   to   any  point   on 

the  joint. 

With  regard  to  Eq.  (4)  it  may  be  stated  that  it  has 
not  heretofore  been  customary  to  consider  the  distribution 
of  shearing  stress  on  horizontal  joints.  But,  if  the  dis- 
tribution of  normal  stresses  may  be  assumed  to  be  repre- 
sented by  Eqs.  (i),  (2),  and  (3),  with  equal  validity  for 
the  usual  types  of  dam,  may  the  shear  at  any  point  be 
assumed  to  be  represented  by  Eq.  (4).  It  is  believed  by 
Mr.  Atcherley  that  these  equations  more  nearly  express  the 
conditions  of  equilibrium  in  a  dam  than  the  usual  ones  do, 
even  though  the  latter  tacitly  assume  the  first  three  by 
imposing  the  condition  of  the  middle  third,  and  use  a  fric- 
tion condition,  instead  of  one  for  shear  as  expressed  by 
Eq.  (4). 

In  reference  to  this  friction  factor  there  may  be  some 
question  of  doubt,  since  M.  Levy*  prescribes  an  angle 
of  30°  for  masonry  on  masonry,  while  Rankine  gives  36°; 
on  the  other  hand,  examination  of  dams  actually  built 
frequently  shows  the  angle  to  lie  somewhere  between  the 
above  values. 

But  whatever  its  exact  value,  the  friction  condition 
leaves  some  doubt  as  to  the  actual  distribution  of  shear 

*"  La  Statique  graphique."  IVe  Partie,  '  Ouvrages  en  Majonne- 
rie,"  page  92. 


172  HIGH    MASONRY    DAM    DESIGN 

over  a  horizontal  joint,  the  variation  of  which  must  be 
known,  in  order  to  determine  the  tensile  and  compressive 
stresses  on  the  vertical  sections  of  the  tail  (i.e.,  downstream 
portion)  of  the  dam.  In  consequence  of  this  the  parabolic 
law  as  expressed  by  Eq.  (4)  has  been  assumed  and  will 
later  be  shown  to  be  more  nearly  correct  than  any  other 
hypothesis. 

According  to  the  author  there  is  no  reason  whatever 
why  dams  should  be  tested  solely  by  taking  horizontal 
cross-sections,  and  asserting  that  the  line  of  resistance 
must  lie  in  the  middle  third,  while  the  stresses  across 
the  vertical  sections  of  the  tail  are  absolutely  neglected. 
If  the  former  condition  is  valid,  then  no  dam  ought  to 
be  passed  unless  it  can  be  shown  also  that  there  is  no  ten- 
sion of  any  serious  value  across  vertical  cross-sections  of 
the  tail,  parallel  to  the  length  of  the  structure.  It  is 
believed  that  a  great  number  of  dams  as  now  designed 
will  be  found  to  have  very  substantial  tension  in  these 
sections  and  this,  in  the  opinion  of  the  author,  is  a  source 
of  weakness  in  dam  construction  which  has  not  been 
properly  considered  and  allowed  for. 

If  the  problem  is  to  be  solved  on  the  assumption  that 
a  dam  is  an  "  isotropic  and  homogeneous  "  structure,  the 
general  equations  for  the  stresses  can  be  determined  only 
by  the  following  considerations: 

(a)  The  normal  and  shearing  stresses  on  the  horizontal 
top   and   curved   flank,   i.e.,   downstream    face,   are   both 
zero. 

(b)  The   normal   stress    on    the   battered  front   or  up- 
stream face  is  equal  to  the  water  pressure,  and  the  shear  is 
zero,  and 


HIGH  MASONRY   DAM  DESIGN  173 

J* 

(c)  Either  the  "stresses  or  the  shifts  must  be  supposed 
given  over  the  base. 

It  follows  at  once  from  this  that  Eqs.  (i),  (2),  and  (3) 
are  not  absolutely  true,  but  that  the  shear  is  fairly  closely 
represented  by  Eq.  (4). 

As  far  as  the  present  investigation  is  concerned,  however, 
the  enquiry  is  not  as  to  the  validity  of  the  usual  treatment; 
it  is  obviously  faulty.  But  it  is  the  purpose  to  try  to 
indicate  that,  supposing  it  to  be  correct,  its  present  partial 
application,  i.e.,  to  horizontal  joints  only,  involves  the 
serious,  and,  it  is  believed,  often  dangerous,  neglect  of 
large  tension  across  the  vertical  sections. 

To  justify  the  above  statement,  two  model  dams  of 
wood  were  employed  for  experimental  purposes,  the  cross- 
sections  being  identical,  and  agreeing  with  that  of  a  dam 
actually  constructed.  One  of  these  models  was  sub- 
divided into  horizontal  strata  to  study  the  effect  on  such 
planes,  and  the  other  into  vertical  longitudinal  strata, 
for  a  similar  purpose.  The  application  of  the  loading 
was  such  that  it  approximated  as  closely  as  possible  the 
conditions  obtaining  in  an  actual  dam.  The  general  con- 
clusions from  these  experiments  were  that: 

(a)  The  current  idea  that  the  critical  sections  of  a  dam 
are    the   horizontal    ones    is    entirely    erroneous.     A    dam 
collapses  first  by  the  tension  on  the  vertical  sections  of 
the  tail. 

(b)  The   shearing   of    the   vertical   sections    over   each 
other  follows  immediately  on  this  opening  up  by  tension. 

(c)  It   is   probable   that   the   shear  on   the   horizontal 
sections  is  also  a  far  more  important  matter  than  is  usually 
supposed. 


174  HIGH    MASONRY    DAM    DESIGN 

It  follows  consequently,  that  keeping  the  line  of  resist- 
ance within  the  middle  third  of  the  horizontal  sections 
is  by  no  means  the  hardest  part  of  dam  design.  It  would 
be  surprising  if,  with  all  the  labor  spent  on  this  point, 
the  bulk  of  existing  dam  constructions  are  not,  for  masonry, 
under  very  considerable  tension,  i.e.,  a  tension  across  the 
vertical  sections  which  has  been  hitherto  disregarded. 

It  is  proposed  therefore  to  lay  it  down  as  a  rule  for  the 
construction  of  future  dams  that  the  stability  of  the  dam 
from  the  standpoint  of  the  vertical  sections  must  be  con- 
sidered in  the  first  place.  If  this  be  satisfactory,  it  is 
believed  that  the  horizontal  sections  will  be  found  to  be 
stable,  but  of  course  the  latter  must  be  independently 
investigated. 

The  above  conclusions  were  apparently  verified  by  a 
combined  analytical  and  graphical  treatment  in  which 
the  algebraical  analysis  will  here  be  considered  first. 

Denoting  the  total  vertical  force  acting  on  a  horizontal 
joint  by  Q0,  and  the  total  horizontal  force  acting  over  the 
same  by  PO,  under  the  assumption  that  the  reservoir  is 
full,  the  variation  of  the  normal  pressure  on  the  joint 
may  be  represented  by  the  straight  line  of  Eq.  (i). 

If  the  resultant  pressure  on  the  joint  be  assumed  to 
cut  it  at  the  extremity  of  the  middle  third,  then  according 

b2 

to  the  previous  notation,  d  will   have  a  value  of  J— ,  pro- 

c 

vided  2b  is  the  length  of  the  joint.  This  indicates  that 
the  line  representing  the  variation  of  normal  pressure 
over  the  joint  intersects  it  at  the  upstream  edge,  and  any 
vertical  between  it  and  the  joint  itself  will  represent  the 
normal  pressure  at  that  point  where  the  vertical  is  erected. 


HIGH   MASONRY   DAM   DESIGN  175 

'  jfc 

Denoting  this  by  yl\i  may  be  termed  "  the  vertical  height- 
giving  pressure,"  and  may  also  be  expressed  in  terms  of 
height  of  masonry,  if  the  factors  upon  which  it  depends 
are  expressed  in  cubic  feet  of  masonry. 

Again,  we  may  write  an  equation  of  the  downstream 
face,  with  respect  to  the  same  joint  so  long  as  that  face 
is  a  straight  line,  by  making  y'  =moc. 

Evidently  then  if  this  latter  line,  and  the  one  indicating 
the  variation  of  pressure  over  the  base,  be  referred  to 
the  same  origin,  the  tip  of  the  tail,  the  difference  in  areas 
included  between  each  and  the  base  will  represent  the 
total  upward  force,  in  cubic  feet  of  masonry,  acting  over 
any  assumed  portion,  "  x"  of  the  joint,  measured  from 
the  tail. 

Representing  this  upward  force  by  FI  its  point  of 
application  may  be  easily  determined,  while  the  shear 
may  be  written  as  F2,  being  regulated  by  Eq.  (4). 

As  FI  and  F%  thus  give  all  the  external  forces,  con- 
sidering a  wedge-shaped  piece  of  dam  bounded  by  the 
downstream  face,  a  vertical  and  a  horizontal  plane,  the 
total  shear  on  the  vertical  plane  must  equal  FI  and  the 
total  thrust  F2,  since  these  internal  stresses  are  held  in 
equilibrium  by  external  forces.  Thus  FI  equals  the  total 
shear  on  the  vertical  section,  at  a  distance  x  from  the 
tip  of  the  tail,  while  F2  equals  the  total  horizontal  thrust 
over  the  same. 

If  y  be  expressed  in  terms  of  x,  and  locate  the  point  on 
the  successive  vertical  planes  through  which  the  resultant 
acts,  then  the  equation  will  represent  the  line  of  resistance 
on  these  vertical  planes.  It  is  found  to  be  an  hyperbola. 

Considering  the  stresses  on  the  vertical  sections,  it  is 


176  HIGH    MASONRY    DAM    DESIGN 

found:  First,  that  the  maximum  shear  may  be  properly 
represented  by  f  the  mean  value,  and  may  be  so  arranged 
as  to  be  expressed  in  terms  of  FI  and  mx.  Such  an  equa- 
tion, representing  a  straight  line,  immediately  shows  the 
necessity  of  thickening  the  tip  of  the  tail  which,  as  a 
matter  of  fact,  is  the  usual  procedure  in  actual  design. 
Second,  the  line  representing  the  maximum  tensile  stress 
may  be  shown  to  vary  as  a  parabola  whose  axis  is  vertical. 
When  the  downstream  face  ceases  to  be  linear,  it 
becomes  necessary  to  apply  a  graphical  solution  for  the 
determination  of  the  stresses.  This  it  is  unnecessary  to 
reproduce  here,  but  the  curves  may  be  said  to  indicate 
the  following  results: 

(1)  That  the  line  of  resistance  for  the  vertical  sections 
lies  outside  the  middle  third  for  rather  more   than  half 
the  vertical  sections.     In  other  words,  these  sections  are 
subjected  to  tension. 

(2)  That  the  tensile  stresses  in  the  tail  are,  for  masonry, 
very  serious,  amounting  to  nearly  10  tons  per  square  foot 
at  the  extreme  tip,  and  to  6  tons  per  square  foot  after  we 
have  passed  the  vertical  section,  where  the  strengthening 
of  the  tail  has  ceased. 

(3)  That  the  maximum  shearing  stresses  amount  to  6 
tons  per  square  foot  at  the  tip  of  the  tail  and  5  tons  per 
square  foot  after  we  have  passed  the  vertical  section,  where 
the  strengthening  of  the  tail  has  ceased.     No  undue  import- 
ance should  be  laid  on  the  actual  values  of  these  "  maxi- 
mum ' '  shears  on  the  vertical  sections  however,  as  they  are 
obtained  from  the  mean  shears  by  using  the  round  multi- 
plier   1.5.     This   round   number   is    assumed    because    the 
maximum  is  certainly  greater  than  the  mean  shear.     The 


HIGH   MASONRY   DAM   DESIGN  177 

> 

actual  distribution4'  of  shear  on  the  vertical  sections  has 
not  been  discussed.  It  could,  of  course,  be  found  from 
that  on  the  horizontal  sections,  if  the  latter  were  really 
known  with  sufficient  accuracy,  by  the  equality  of  the 
shears  on  two  planes  at  right  angles.  It  is  sufficient  to 
show  that  the  mean  shears  on  the  vertical  sections  appear 
to  be  higher  than  those  on  the  horizontal  section,  and 
thus  indicate  that  the  parabolic  distribution  applied  to 
sections  some  way  above  the  base,  probably  under-esti- 
mates  the  max'mum  shearing  in  the  dam. " 

In  other  words:  Whether  the  test  is  made  by  the  line 
of  resistance  lying  outside  the  middle  third,  or  by  the  ex- 
istence of  serious  tensile  stresses,  or  by  the  magnitude  of 
the  mean  shearing  stresses,  the  vertical  sections  are  critical 
for  the  stability  in  a  far  higher  degree  than  the  horizontal 
sections. 

In  a  well-designed  dam,  all  the  conditions  for  stability 
of  the  horizontal  sections  may  have  been  satisfied,  yet 
if  the  very  same  conditions  be  applied  to  the  vertical 
sections  not  one  of  them  will  be  found  to  be  satisfied. 
It  seems  accordingly  very  unsatisfactory  that  the  current 
tests  for  stability  should,  if  they  are  legitimate,  be  applied 
to  the  horizontal  instead  of  to  the  far  more  critical  vertical 
sections.  In  the  case  of  the  latter  they  fail  completely; 
and  if  higher  tension  and  shear  are  to  be  allowed  in  the 
vertical  sections,  then  it  is  absurd  to  exclude  them  in  the 
case  of  the  horizontal  sections..  It  is  maintained  by  the 
author  that  the  current  treatment  of  dams  is  fallacious, 
for  it  screens  entirely  the  real  source  of  weakness,  namely, 
in  the  first  place  the  tension,  and  in  the  second  place  the 
substantial  shear,  in  the  vertical  sections,  and  this  at  dis- 


178  HIGH    MASONRY    DAM     DESIGN 

tances  from  the  tail  far  beyond  the  usual  tail-strengthen- 
ing range. 

Nor  do  these  theoretical  results  stand  unverified  by 
experiment;  they  are  absolutely  in  accord  with  the  ex- 
periments on  the  model  dams.  These  collapsed  precisely 
as  might  have  been  expected  from  the  above  investiga- 
tion, i.e.,  the  dam  with  vertical  sections  gave  long  before 
the  dam  with  horizontal  sections.  The  former  collapsed 
by  opening  up  of  the  joints  by  tension  towards  the  tail, 
followed  almost  immediately  by  a  shear  of  the  whole 
structure.  In  the  case  of  the  horizontally  stratified  dam, 
the  collapse,  which  occurred  much  later,  was  by  shear  of 
the  base,  followed  almost  simultaneously  by  a  shear  of 
one  or  more  of  the  horizontal  sections. 

The  question  then  arises  as  to  how  far  the  previously 
assumed  distribution  of  shear  affects  the  main  features 
of  the  results,  and  so  the  other  extreme  was  taken,  i.e., 
uniform  shear,  and  the  effect  determined. 

This  distribution  must  be  further  from  the  actual 
than  the  first  hypothesis,  yet  it  is  still  found: 

(1)  That  the  line  of  resistance  falls  well  outside  the 
middle  third  for  about  half  the  dam. 

(2)  That  there  exist  considerable  tensions,  3  to  4  tons 
per  square  foot,  in  the  masonry. 

(3)  That  the  average  shearing  stresses  on  the  vertical 
sections  are  greater  than  on  the  horizontal  sections.     As  a 
result  of  this  extreme  case,   it  is   believed  that  the  real 
distribution  of  shear  over  the  base,  whatever  it  may  be, 
must  lead  us  to  a  line  of  resistance  lying  well  outside  the 
middle   third,    and   to    tensions   amounting    to    something 
between  5  and  10  tons  per  square  foot. 


HIGH   MASONRY   DAM  DESIGN  179 

> 

From    these    investigations    the    author    concludes    as 
follows : 

(1)  The  current  theory  of  the  stability  of  dams  is  both 
theoretically  and  experimentally  erroneous,  because : 

(a)  Theory  shows  that  the  vertical  and  not  the  hori- 
zontal sections  are  the  critical  sections. 

(b)  Experiment  shows  that  a  dam  first  gives  by  tension 
of  the  vertical  sections  near  the  tail. 

(2)  An  accepted  form  of  cross-section  is  shown  to  be 
stable  as  far  as  the  horizontal  sections  are  concerned,  but 
unstable    by    applying    the    same    conditions    of    stability 
to  the  vertical  sections. 

(3)  The  distribution  of  shear  over  the  base  must  be 
more  nearly  parabolic  than  uniform,   but  as  no  reversal 
of  the  statements  follows  in  passing  from  the  former  to 
the  latter  extreme  hypothesis,   it  is  not  unreasonable  to 
assume  the  former  distribution  will  describe  fairly  closely 
the  facts  until  we  have  greater  knowledge. 

(4)  In  future  it  is  held  that  in  the  first  place  masonry 
dams  must  be  investigated  for  the  stability  of  their  vertical 
sections.     If  this  be  done  it  is  believed  that  most  existing 
dams   will   be   found   to   fail,    if   the   criteria   of   stability 
usually  adopted  for  their  horizontal  sections  be  accepted. 
This  failure  can  be  met  in  two  ways : 

(a)  By  a  modification  of  the  customary  cross-section. 
It  is  probable  that  a  cross-section  like  that  of  the  Vyrnwy 
dam  would  give  better  results  than  more  usual  forms. 

(b)  By  a  frank  acceptance  that  masonry,  if  carefully 
built,  may  be  trusted  to  stand  a  definite  amount  of  tensile 
stress.     It  is  perfectly  idle  to  assert  that  it  is  absolutely 
necessary  that  the  line  of  resistance  shall  lie  in  the  middle 


180 


HIGH    MASONRY    DAM    DESIGN 


third  for  a  horizontal  treatment,  when  it  lies  well  out- 
side the  middle  third  for  at  least  half  the  dam  for  a 
vertical  treatment. 

Immediately  upon  the  publication  of  the  preceding 
results,  Sir  Benjamin  Baker  undertook  some  experiments 
of  a  like  nature.*  The  models  employed  by  him  were 


Radius  =Infinity 


__.*._ 


117175 
Vyrnwy  Dam 

FlG.  29. 


made  of  ordinary  jelly  however,  and  included  not  only 
the  transverse  section  of  the  dam  itself,  but  the  rock  upon 
which  it  rested  as  well.  It  is  shown  in  the  figure. 

The  horizontal  and  vertical  lines  drawn  on  the  sides 
of  the  model  were  for  the  purpose  of  detecting  any  dis- 
tortion that  might  result  through  the  application  of 

*  Vol.  162,  page  120.     Minutes  of  Proceedings  of  the  Institution  of 
Civil  Engineers. 


HIGH  MASQNRY    DAM   DESIGN 


181 


pressure.  These  pressures  were  applied  against  both 
the  upstream  face  and  the  floor  of  the  reservoir,  as  it 
was  believed  that,  while  according  to  the  theory  of  the 
middle  third  there  could  be  no  tension  in  the  heel,  never- 
theless for  the  case  of  reservoir  full,  fairly  severe  tension 
in  the  masonry  might  thus  be  caused. 

The    experiments    indicated    that    the    distribution    of 
shearing  stress  in  the  plane  of  the  base,  i.e.,  where  the 


FIG.  30. 

dam  met  the  rock,  was  more  nearly  uniform  than  para- 
bolic, and  that  the  strain  extended  into  the  rock  for  a 
distance  equal  to  about  half  the  height  of  the  dam  before 
it  became  undetectable.  To  solve  the  complete  problem, 
therefore,  it  would  be  necessary  to  consider  the  elasticity 
of  the  rock  on  which  the  dam  rested.  Partially  as  a  result 
of  these  and  the  previous  experiments,  it  may  be  pointed 
out  in  passing,  the  proposed  increase  in  elevation  of  the 
Assouan  dam,  whereby  the  capacity  of  the  reservoir 


182  HIGH    MASONRY    DAM    DESIGN 

would  have  been  considerably  augmented,  was  indefinitely 
postponed. 

The  new  feature  in  Atcherley's  analysis  is  that,  even 
though  the  condition  of  "  no  tension  in  a  horizontal 
joint  "  is  satisfied,  dangerous  tensions  may  be  shown  to 
exist  across  vertical  planes.*  In  connection  with  this 
consider,  for  example,  a  section  of  the  dam  ABC,  which 
is  triangular  in  profile,  and  construct  EEC  so  that  the 
ordinates  represent  the  variation  of  the  unit  normal 
stress  over  the  horizontal  joint  BC. 

Taking  a  vertical  section  IK  in  which  /  locates  the 
centroid,  the  forces  to  the  left  are  the  upward  pressure 
acting  over  BK,  tending  to  cause  rota- 
tion in  a  clock-wise  manner  and  thus 
produce  tension  at  K,  and  two  counter- 
acting forces  tending  to  neutralize  this 
pressure :  the  weight  of  the  portion  BKI 
and  the  horizontal  shearing  force  acting 
along  BK.  The  resultant  effect  of  all 
three  will  be  tension  at  K,  provided  the 
FIG.  31.  rotation  is  right-handed,  with  a  conse- 

quent splitting  along  the  vertical  plane  IK. 

In  view  of  the  fact  that  the  horizontal  shear  is  present 
as  a  factor,  it  is  necessary  to  determine  its  distribution, 
and  this  Prof.  W.  C.  Unwin  undertook  to  do.t  Instead 
however,  of  accepting  the  distribution  in  accordance  with 
Atcherley's  assumptions,  an  analysis  was  attempted  by 


*"  Engineering,"  Vol.  79,  page  414. 

t"  Engineering,"  Vol.  79,  page  513.     "Note  on  the  Theory  of  Un- 
symmetrical  Masonry  Dams,"  by  W.  C.  Unwin. 


HIGH   MASONRY   DAM   DESIGN 
i 


183 


which  the  shear  might  be  actually  calculated,  and  in 
doing  so  attention  was  called  to  the  fact  that  the  accepted 
theory  of  dam  design  is  incomplete  in  just  that  feature, 
since  it  fails  to  consider  the  rate  of  change  in  the  hori- 
zontal shear. 

In  any  analysis  the  fundamental  assumption  must  be 
made  that  a  masonry  dam  is  a  homogeneous-elastic  solid, 
and,  while  it  is  not  absolutely  essential  that  no  tension 
exist  at  any  point  in  the  cross-section,  yet  it  seems  desir- 
able that  there  should  be  none  at  the  upstream  face  of 
horizontal  joints. 

It  may  be  said  therefore,  that  for  a  more  exact  analysis 
the  problem  resolves  itself  into  one  of  the  determination 
of  shear  on  horizontal  planes,  and  Prof.  Unwin  suggests 
as  follows,  a  method  of  procedure  by  which  this  may  be 
accomplished : 

If,  as  in  the  figure,  we  assume  a  dam  of  triangular 
section,  in  which  AB  is  some 
horizontal  joint,  other  than 
the  base,  and  C  its  centroid, 
then  Q  will  represent  the  water 
thrust,  P  the  weight  of  ma- 
sonry, and  R  their  resultant. 

In  agreement  with  the  or- 
dinary theory  we  may  write 
the  well-known  formula  for 
the  unit  normal  pressure  on  a 
horizontal  joint,  at  any  point  x,  measured  from  A,  as 
follows : 


*_.. 


*-$+*%&} 


(5) 


184 


HIGH   MASONRY    DAM   DESIGN 


For  the  horizontal  shear  we  must  proceed  further. 
Consider,  therefore,  the  forces  to  the  left  of  HK  in  Fig.  33 , 
we  have  (i)  the  vertical  pressure  on  AK,  (2)  the  weight 
of  AHK,  and  (3)  the  shear  acting  along  AK.  It  is 
evident  that  the  difference  between  (i)  and  (2)  repre- 
sents the  total  vertical  shear  on  HK. 

If,  therefore,  the  figure  ALMB  represent,  in  masonry 
units,  the  distribution  of  normal  stress  on  A  B,  as  given 
by  Eq.  (i),  then  ALTH  will,  in  like  manner,  represent 
the  above-mentioned  total  vertical  shear  on  HK. 


B' 


FIG.  33- 


FIG.  34- 


Consider  now  a  second  section  A  'B',  a  small  distance 
z  above  AB\  the  total  shear  on  HK'  may  then  be  found  as 
before.  Denoting  the  former  by  5,  and  the  latter  by 
5',  then  S-Sf  equals  the  total  shear  on  KK' ',  which, 
when  divided  by  zt  will  give  the  intensity  of  vertical  shear 
at  K,  and  consequently  the  intensity  of  horizontal  shear 
at  the  same  point. 

•  Since  all  the  forces  to  the  left  of  HK  are  now  known, 
the  normal  stress  on  that  plane  may  be  found,  and  from 
it  we  may  readily  determine  whether  tension  or  com- 
pression exists  at  K. 

At  the  base  these  results  would   be  much   modified, 


HIGH   M;\£ONRY    DAM    DESIGN 


185 


because  of  the  discontinuity  of  form,  which,  in  the  opinion 
of  Prof.  Unwin,  places  the  exact  determination  of  the 
stresses  beyond  the  power  of  mathematics.  The  author 
believes  the  effect  of  the  rock  into  which  the  dam  is  built 
is  to  reduce  the  variation  of  stress  which  would  otherwise 
exist. 

In  a  subsequent  paper,*  giving  a  complete  demon- 
stration of  the  preceding  analysis  as  applied  to  a  masonry 
dam  of  triangular  cross-section,  it  is  found  that  the  dis- 
tribution of  shear  on  a  plane  horizontal  joint  may  be 
represented  by  a  right  triangle  whose  base  is  the  length  of 
the  joint  and  whose  vertex  is 
perpendicularly  below  the  down- 
stream edge.  The  figure  illus- 
trates the  variation  of  normal 
stress  and  shear  on  AB\  the 
lines  of  resistance  for  both 
vertical  and  horizontal  planes; 
and  the  centers  of  gravity  of 
the  sections  above  the  successive 
horizontal  joints. 

Consequently  the  total  nor- 
mal or  shearing  stress  on  any 
part  of  AB  is  equal  to  the  area 
between  that  part  and  the  line 

of  normal  stress  or  the  line  of 

FIG.  35. 

shearing  stress.. 

If  the  upward  reactions  and  the  weights  of  the  dam  to 


*"  Engineering,"  Vol.  79,  page  593.     "Further  Note  on  the  Theory 
of  Unsymmetrical   Masonry  Dams."     W.  C.  Unwin. 


186  HIGH    MASONRY    DAM    DESIGN 

the  left  of  each  vertical  section  be  combined  with  the 
shears  T,  acting  along  AB,  the  resultants  will  cut  the 
vertical  sections  at  points  shown  on  the  line  of  resist- 
ance for  these  vertical  sections.  As  this  line  lies  wholly 
within  the  middle  third,  there  can  be  no  tension  on  any 
vertical  section. 

The   total  compressive  stress   on  any  vertical  section 
at  its  lower  edge  will  therefore  be: 


where  T  is  the  shear  on  the  horizontal  plane  from  the  toe 
to  the  vertical  section  taken,  y  the  height  of  the  vertical 
section,  and  z  the  distance  from  the  center  of  the  vertical 
section  to  the  point  of  application  of  the  resultant  forces 
on  that  section. 

Near  the  upstream  toe  the  plane  on  which  the  greater 
principal  stress  acts  is  found  to  be  vertical  while  near 
the  downstream  toe  it  approaches  the  horizontal.  The 
stresses  are  all  compressive  and  on  the  water  face  the 
compressive  stress  is  at  all  points  equal  to  the  water 
pressure  at  that  point. 

The  above  analysis  is  simply  an  application  to  vertical 
sections  of  the  method  now  accepted  as  applicable  to  the 
horizontal  planes  and  is  a  possible  solution,  since  the 
distribution  of  shear  is  known.  It  differs  from  Atcherly's 
method  in  the  fact  that  the  latter  assumes  the  usual 
distribution  of  normal  stress,  together  with  a  parabolic 
variation  for  the  horizontal  shear.  This  latter  hypothesis 
the  author  thinks  inconsistent  with  the  previous  one. 


HIGH  MASONRY    DAM    DESIGN  187 


Further  investigations  ^y  Prof.  Unwin*  on  dams  of 
various  sections  lead  to  the  following  conclusions: 

(1)  For  a  rectangular  dam  the  distribution  of  shearing 
stress   on  horizontal   planes   may   be   represented   by   the 
ordinates  of  a  parabola. 

(2)  For   a    triangular   dam,    the    distribution    may    be 
represented  by  the  ordinates  of  a  triangle  with  the  apex 
below  the  downstream  toe. 

(3)  For  a  dam  with  vertical  upstream  face  and  curved 
downstream    face    the    distribution    may    be    represented 
by    a    figure   consisting   of    a    parabola    superposed   on   a 
triangle. 

(4)  For  a  dam  with  rectangular  base  the  distribution 
is  represented  by  a  parabola. 

Following  the  results  of  the  experimental  investigations 
of  Atcherley  and  Baker,  several  other  papers  of  a  like 
nature  appeared  in  the  Minutes  of  Proceedings  of  the 
Institute  of  Civil  Engineers,  Vol.  162.  The  first  of 
these  to  be  considered  here  is  that  by  Sir  John  Walter 
Ottley  and  Arthur  William  Brightmore,  entitled,  "Ex- 
perimental Investigations  of  the  Stresses  in  Masonry  Dams 
subjected  to  Water- Pressure." 

In  presenting  this  paper,  the  authors  drew  attention 
to  the  fact  that  until  the  publication  of  Mr.  Atcherley's 
results,  the  question  of  dam  design  had  been  accepted 
as  settled,  and  that  his  memoir  had  had  the  effect  of 
reopening  the  entire  subject  of  the  distribution  of  stress 
in  structures  of  this  class. 

*" Engineering, "  Vol.  79,  page  825.  "On  the  Distribution  of 
Shearing  Stress  in  Masonry  Dams."  Prof.  W.  C.  Unwin. 


188  HIGH  MASONRY   DAN   DESIGN 

It  was  also  pointed  out  that  tension  was  found  by  him 
to  exist  on  vertical  planes  near  the  outer  toe,  whether 
the  distribution  of  shearing  stress  over  the  base  was 
assumed  to  be  uniform  or  to  vary  according  to  the  para- 
bolic law. 

Considering  a  transverse  section  of  a  dam,  the  authors 
argued  that,  whatever  the  distribution  of  shear  over  the 
base  might  be,  it  must  follow  some  other  law  near  the  top, 
since  the  conditions  in  these  higher  levels  are  radically 
different  from  those  existing  in  the  lower,  where  the  dam 
is  fixed  to  the  foundation,  and  where  the  water  pressure 
ceases  abruptly. 

The  investigation  was  therefore  undertaken,  at  least 
in  part,  to  determine  the  distribution  of  shear  on  horizontal 
planes  in  the  higher  levels  of  the  dam  and  to  see  how  it 
varied  from  that  at  the  base;  and  it  might  be  stated  here 
that  it  was  found  to  be  uniform  in  the  latter  plane  but  to 
vary  uniformly  from  zero  at  the  heel  to  a  maximum  at 
the  toe  in  the  higher  levels,  the  change  from  the  one 
condition  to  the  other  being  gradual.  It  will  be  shown 
that  it  is  near  the  inner  toe  rather  than  near  the  outer 
toe  that  tension  may  be  anticipated. 

The  model  dams  were  triangular  in  section,  made  from 
a  kind  of  modeling  clay  called  "  plasticine,"  and  so  pro- 
portioned that  the  resultant  pressure  on  the  base  cut 
that  plane  at  the  downstream  extremity  of  the  middle 
third. 

For  purposes  of  observation  the  sections  were  placed 
between  vertical  sides  of  plate  glass,  upon  which  vertical 
and  horizontal  lines  had  been  etched,  corresponding  to 
similar  lines  on  the  model,  so  that  any  displacement  in 


HIGH    MASONRY    DAM    DESIGN  189 

& 

the  latter  might  fre  noted  by  comparison  with  the  former. 
Pressure  was  applied,  by  means  of  a  thin  rubber  bag  con- 
taining water  which  was  made  to  fit  the  frame.  Though 
the  water  was  allowed  to  act  over  a  period  of  33  days, 
after  the  elapse  of  one  week  a  crack  was  noticed  at  the 
upstream  toe,  running  downward  and  at  an  angle  of 
about  45°.  At  the  end  of  the  longer  period  an  examination 
showed  that  in  the  neighborhood  of  the  base  the  dis- 
placement of  the  vertical  lines  was  such  as  to  make  them 
all  about  equally  inclined,  thus  indicating  a  uniform 
intensity  of  shear  on  that  section,  while  in  the  higher 
levels  and  near  the  outer  portion  of  the  dam  the  lines 
became  more  inclined  as  the  elevation  increased,  indicating 
that  the  intensity  of  shear  increased  also  as  the  top  was 
approached. 

Turning  to  the  horizontal  lines  in  the  model  for  the 
purpose  of  discovering  the  method  of  distribution  of 
normal  stress,  it  was  found  that  they  were  curves  at  the 
base,  sloping  downward  from  the  inner  toe  to  a  point 
about  two-thirds  the  distance  to  the  outer  toe,  then  re- 
maining fairly  level  until  almost  reaching  the  down- 
stream face,  when  they  finally  bent  up  slightly.  In  the 
higher  levels,  however,  these  lines  gradually  developed  a 
uniform  slope  running  from  the  inner  to  the  outer  toe. 

An  investigation  of  the  shearing  stresses  on  vertical 
planes  requires  that,  to  draw  the  line  representing  the 
intensity  of  normal  reaction  at  the  base  the  following 
facts  must  be  considered: 

(1)  The  total  normal  reaction  equals  the  weight  of  the 
dam. 

(2)  Since  the  resultant  pressure  on  the  base  acts  at 


190 


HIGH   MASONRY    DAM    DESIGN 


one-third  the  width  from  the  outer  toe,  the  moment  of  the 
reaction  stresses  about  this  point  must  be  zero. 

(3)  The  intensity  of  the  reaction  at  the  outer  toe 
must  equal  the  intensity  of  the  shearing  stress  in  the  vertical 
plane  multiplied  by  the  ratio  of  the  height  to  the  base  of 
the  dam. 


FIG.  36. 

Referring  to  the  figure:  AB  represents  the  base  of  the 
dam,  and  BC  twice  the  average  intensity  of  normal  stress 
on  AB.  AC  is  then  drawn;  consequently  ABC  represents 
the  total  normal  stress  on  A  B,  or  the  weight  of  the  structure. 


HIGH    MASONRY   DAM    DESIGN  191 

i 

If  AE,  on  the  other  hand,  represents  the  actual  intensity 
of  normal  reaction  over  AB,  then  for  (i)  to  hold  true  the 
area  Y  must  equal  the  areas  (x  +  z)  and  if  (2)  is  to  hold, 
the  moments  of  x,  y,  and  z,  about  D  (equal  to  \AB  from 
B),  must  be  zero;  also  for  (3)  to  be  satisfied,  BE  must 
equal  the  limiting  value  of  shearing  stress  in  a  vertical 
plane  near  the  toe,  multiplied  by  the  height  and  divided 
by  the  base  of  the  dam. 

From  these  considerations  AE  may  be  fitted  in  by 
trial  till  it  is  found  to  satisfy  all  of  the  above  conditions. 

Dividing  the  cross-sections  into  vertical  strips  i  inch 
wide  we  may  properly  consider  the  equilibrium  of  each  such 
strip.  Evidently  the  difference  between  the  weight  of 
each  strip  and  the  normal  reaction  on  the  base  is  equal 
to  the  difference  in  shear  on  the  two  adjacent  vertical 
planes,  and  if  in  the  figure  these  weights  be  plotted  upward 
from  AE,  the  curve  FE  will  result.  Furthermore,  both 
the  curves  for  "  total  shear  on  vertical  planes  "  and 
"  average  intensity  of  shear  on  vertical  planes  "  may  now 
be  drawn,  whereupon  it  is  evident  to  what  extent  the 
average  intensity  of  shear  on  vertical  planes  varies,  and 
how  it  compares  with  the  average  intensity  on  the 
base. 

Since  the  shear  on  horizontal  and  vertical  planes  at 
any  one  point  is  equal,  and  the  shear  on  the  base  is  practi- 
cally constant,  it  follows  that  above  the  base  the  shear  on 
horizontal  or  vertical  planes  is  small  near  the  heel  while 
in  the  outer  half  above  the  base  it  increases  as  the  outer 
edge  is  approached;  in  fact  it  increases  from  zero  at  the 
heel  to  a  maximum  at  the  toe.  These  facts  show  that  the 
shearing  stresses  to  be  provided  for  are  those  existing 


192  HIGH    MASONRY    DAM    DESIGN 

in  the  higher  levels  and  near  the  toe,  and  not  those  at  the 
base. 

In  considering  the  effect  of  shear  on  the  base,  neglecting 
the  "  fixing  "  at  that  level,  we  may  assume  that  the  re- 
action stress  and  that  due  to  the  weight  of  a  strip,  is  constant 
over  each  inch.  They  then  act  at  the  middle  of  each 
strip;  and,  taking  these  points  successively  as  centers, 
the  difference  of  the  moments  of  the  horizontal  pressures 
on  the  vertical  sides  of  the  strip,  it  is  evident,  will  equal 
the  sum  of  the  shearing  stresses  on  the  same  vertical  sides 
multiplied  by  \  inch. 

This  makes  possible  the  determination  of  the  moment 
of  the  horizontal  pressures  on  each  vertical  strip. 

The  horizontal  shear  on  each  inch  of  base  being  the 
difference  between  the  horizontal  pressures  acting  on  the 
two  vertical  sides,  the  latter  may  be  determined  as  soon 
as  their  points  of  application  are  given.  As  these  points 
are  known  for  the  innermost  and  outermost  strip,  an  easy 
curve  may  be  drawn  which  will  approximately  locate  the 
other  points  and  thus  give  the  desired  heights.  From 
these  results  it  may  be  shown  that  the  shearing  stress  on 
the  base  increases  from  practically  zero  at  the  inner  toe 
to  a  point  near  the  center  of  the  base  and  then  remains 
fairly  constant. 

The  modification  of  this  distribution,  due  to  the  fixing 
of  the  dam  to  its  base,  must,  on  the  other  hand,  be  con- 
sidered. The  water  tends  to  cause  a  maximum  pressure 
and  displacement  at  the  inner  face,  which  diminishes  to 
zero  at  the  outer.  As  the  dam  is  fixed,  this  displacement 
is  prevented,  thus  inducing  corresponding  shears,  and  the 
effect  of  this  conflicting  condition,  with  that  previously 


HIGH   MASONRY   DAM    DESIGN  193 

j 

shown  to  exist,  causes  a"*nearly  uniform  shear  over  the 
base. 

Further  evidence  of  uniform  shear  on  the  base  was 
obtained  as  follows:  The  models,  after  being  subjected 
to  water  pressure,  showed  cracks  which  appeared  at  the 
inner  toe,  the  angles  which  these  made  with  the  horizontal 
steadily  diminishing  as  the  base  was  decreased  in  width 
from  a  maximum  of  45°  for  the  widest  base  used  to  25° 
for  the  narrowest. 

The  variation  of  these  inclinations  corresponded  closely 
with  the  computed  directions,  on  the  assumption  that  the 
shear  was  uniform  over  the  base  and  the  experiments 
therefore  strongly  support  the  inference  that  shear  over 
the  base  is  uniformly  distributed. 

It  was  shown  by  means  of  the  models  that  there  are 
tensile  stresses  on  other  than  horizontal  planes  passing 
through  the  inner  toe.  The  models  indicated  this  by 
cracking,  even  when  the  back  was  sloped  away  from  the 
vertical  so  as  to  cause  vertical  pressure  and  hence  com- 
pression on  the  upstream  face. 

The  impossibility  of  tension  on  vertical  planes  near 
the  outer  toe  may  be  shown  by  means  of  the  following 
equation  for  principal  stress: 


r     r    '  r   —    '   \r    i  r  /         *t\rr       **  /  /    \ 

where  compressions  are  plus  and  tensions  are  minus. 
When  pp'  >  q2  at  any  point,  there  can  be  no  tension  at 
that  point,  since  under  the  above  conditions  both  principal 
stresses  will  be  compression  and  hence  stresses  on  all  other 
planes  passing  through  that  poinf  will  be  compression 


194  HIGH    MASONRY    DAM    DESIGN 

also.  This  condition  may  be  shown  to  exist  near  the  outer 
toe,  and  hence  no  tension  can  act  across  any  vertical 
plane  in  that  position. 

-  For  example  consider  the  equilibrium  of  a  wedge  of 
unit  length  cut  off  by  a  vertical  plane  near  the  toe. 

p'  =  intensity  of  pressure  normal  to  a  vertical  plane 

at  base. 

p  =  intensity  of  reaction  normal  to  the  base. 
q  =  intensity  of  shearing  stress. 

Then  p'h  =  qb  or, 


(8) 


The  weight  of  the  particle  is  negligible  because  it  varies 
with  h2. 

Since  the  resultant  stress  must  be  parallel  to  the  outer 
face  it  follows  that, 


(9) 


Multiplying  (8)  by  (9)  there  results 

pp'  =q2  at  outer  toe.  But  p  has  been  shown  to  in- 
crease for  some  distance  from  outer  toe  and  the  point  of 
application  of  pf  becomes  relatively  lower  as  the  inner 
toe  is  approached  and  since  the  average  pressure  is  con- 
stant it  follows  that  p  increases  as  the  distance  from  the 
outer  toe  increases  and  hence  in  the  vicinity  of  the  outer 
toe  ppf  is  greater  than  <f  and  consequently  there  can  be 
no  tension  in  that  neighborhood.  This  was  checked  by 
the  behavior  of  the  models. 


HIGH    MASONRY    DAM    DESIGN  195 

P 

It  is  a  fact  that  in  dam  work  the  normal  stress  is  the 
only  one  specified,  whereas  the  absolute  maximum  is  about 
50  per  cent  greater. 

The  conclusions  reached  from  this  set  of  experiments 
follow  : 

(1)  If  a  masonry  dam  be  designed  on  the  assumption 
that  the  stresses  on  the  base  are  uniformly  varying  and 
that  the  stresses  are  parallel  to  the  resultant  force  acting 
on  the  base,  the  actual  normal  and  shearing  stresses  on 
both  horizontal  and   vertical  planes  would  be  less   than 
those  provided  for. 

(2)  There  can  be  no  tension  on  any  planes  near  the 
outer  toe. 

(3)  There  will  be  tension  on  certain  planes  other  than 
the    horizontal    near    the    inner    toe,    and    the    maximum 
intensity  of  such  tension  in  the  foundation  being  generally 
equal  to  the  average  intensity  of  shearing  stress  on  the 
base,  and  the  inclination  of  its  plane  of  action  being  about 
45°;   and  its  maximum  intensity  in  the  dam  above  the  base 
about  J  the  above  amount  and  acting  on  a  plane  less  in- 
clined to  the  horizontal. 

The  investigation  undertaken  by  Mr.  Hill  *  for  "  The 
Determination  of  the  Stresses  on  any  Small  Element  of 
Mass  in  a  Masonry  Dam,"  are  on  the  other  hand  purely 
analytical  in  character,  being  directed  toward  a  solution 
of  (i)  the  vertical,  (2)  horizontal,  and  (3)  tangential  shearing 
forces  acting  on  the  faces  and  along  the  edges  of  such  an 
element. 

*  Minutes  of  Proceedings  of  the  Inst.  of  C.  E.,  72. 


196  HIGH    MASONRY   DAM   DESIGN 

In  this  analysis,  there  is  first  expressed  a  perfectly 
general  formula  for  C  (the  distance  of  the  load  point 
from  the  center  of  the  joint),  and  two  other  general  formulae 
for  the  pressures  p\  and  p2  in  terms  of  the  total  load  and 
C  from  its  above  value,  where  pi  is  the  minimum  and 
p2  the  maximum  pressure.  For  the  pressure  p  at  any 
point  x  on  the  joint  of  length  b  the  following  equation 
is  used: 


(10) 


Up  to  this  point  the  analysis  is  identical  with  the  general 
procedure  of  investigation,  which  assumes  that  the  hori- 
zontal pressures  are  proportional  to  the  vertical,  and 
does  not  analyze  the  shear. 

Citing  Prof.  Unwin,  the  author  states  tnat  the  former 
"  suggested  that  the  shearing  stress  at  any  point  might 
be  found  by  considering  the  difference  between  the  total 
net  vertical  reactions  (between  that  point  and  either  face) 
along  two  horizontal  planes  a  unit's  distance  apart,  and 
has  applied  the  principle  by  the  use  of  algebraical  methods." 
Mr.  Hill,  on  the  contrary,  employs  the  calculus  to  obtain 
more  rigorous  results. 

The  procedure  follows:  Consider  any  point  distant  x 
from  the  inner  toe  and  on  the  lower  of  two  horizontal 
planes,  a  unit's  distance  apart.  The  total  vertical  reaction 


is  then    I    pdx.     Subtracting  the  weight  of  masonry  resting 
JQ 

on  this  portion  of  the  horizontal  joint,  and  denoting  the 
difference  by  r  we  have  an  expression  for  the  "  net 
vertical  reaction."  If  this  value  of  r  be  differentiated 


HIGH   MASONRY   DAM   DESIGN  197 

—  '  jit 

with  respect  to  K,  the  distance  between  the  two  hori- 
zontal planes,  the  change  in  the  reaction  will  be  obtained, 
and  this  change  or  difference  is  the  vertical  shearing 
stress  at  the  point  located  by  x.  It  is  also,  therefore,  the 
horizontal  shear  at  the  same  point,  which  we  may  denote 
byq. 

If  q  be  integrated  with  respect  to  x,  between  the  limits 
of  x  and  b,  the  resulting  expression  will  give  the  entire 
horizontal  shear  between  such  limits  on  the  joints  in 
question.  Represent  this  by  Qx. 

To  find  the  horizontal  pressure  intensity,  we  have 
but  to  consider  the  above  integration.  This  shear  must 
be  resisted  by  the  material  along  the  vertical  section  at  x. 
Similarly  the  total  shear  on  a  plane  a  differential  distance 
below  the  last  must  be  resisted  by  the  vertical  section 
at  x,  differing  in  height  from  the  former  by  dh.  Conse- 
quently the  differential  of  Qx  with  respect  to  h=p'  will 
represent  the  horizontal  pressure  intensity  at  point  x. 
These  expressions  for  p,  pf  and  q  therefore  give  respectively 
the  values  of  the  vertical  pressure  intensity,  horizontal 
pressure  intensity,  and  shearing  force  acting  on  a  unit 
element  of  mass. 

Cain  *  presents  a  treatment  of  this  matter,  which, 
while  presenting  no  new  features,  is  strictly  arithmetical 
in  character,  and  in  that  respect  at  least  differs  from  the 
preceding.  Its  purpose,  as  Hill's,  is  to  determine  the 
amount  and  distribution  of  stress  at  any  point  in  a  masonry 


*  Wm.  Cain,  M.  Am.  Soc.  C.  E.,   Trans.   Am.   Soc.  C.    E.,   Vol.  64, 
page  208. 


198  HIGH    MASONRY    DAM    DESIGN 

dam,   on  the  assumption   that   the  law  of  the  trapezoid 
represents  the  variation  of  pressure  on  horizontal  joints. 

The  analysis  finally  establishes  formulae  for  (i)  the 
normal  unit  stress  at  any  point  in  a  horizontal  joint,  (2) 
the  normal  unit  stress  on  a  vertical  plane  at  any  point 
of  a  horizontal  joint,  (3)  the  unit  shear  on  either  horizontal 
or  vertical  planes  at  any  point  of  a  horizontal  joint,  and 
at  the  same  time  indicates  the  method  of  determining  the 
maximum  and  minimum  normal  stresses  and  the  planes 
on  which  they  act. 

The  solutions  are  only  approximate,  but  the  results  are 
found  to  be  close  enough  for  the  purpose. 

Before  proceeding  it  may  be  advisable  to  review  certain 
features  involved  in  a  consideration  of  the  stresses  in  a 
masonry  dam  which  Prof.  Cain  presents  in  a  very  satis- 
factory manner. 

i.  It  will  be  evident  from  an  examination  of  the  figure 
that  the  intensities  of  shear  on  two  planes  at  right  angles 

to  each  other  are  equal.  For,  in 
the  elementary  cube  under  consider- 
ation, the  weight  may  be  neglected, 
-  since  it  is  an  infinitesimal  of  the  third 
order,  while  the  opposing  normal 
forces  balance  as  the  cube  is  reduced 
in  size. 

For  equilibrium  then,  q-a-a  = 
q'  -a-a,  or  q  =  qf  and,  because  each  side  is  a  differential 
quantity,  it  may  be  assumed  that  the  values  q  and  q' 
represent  the  average  unit  shear  on  the  respective  faces. 
As  a  consequence  they  are  equal  to  the  shear  at  any 
point,  for  example  A,  of  the  particle. 


HIGH    MASONRY   DAM    DESIGN 


199 


2.  In  a  triangular  element  of  the  dam,  Fig.  38,  at  the 
down-stream  edge,  and  of  unit's  length,  the  forces  acting 
are  those  shown.  Because  it  is  an  element  we  may  neglect 
the  weight,  and  therefore,  if  pr  is  the  normal  intensity  of 
stress  on  a  vertical  plane,  p  the  normal  intensity  of  stress 
on  a  horizontal  plane,  and  q  the  shear  intensity,  for 


for 


then 


or, 


or     p—q~r     and     q=ptd.n(f).    .     (n) 
o 


I.H  =  o,p'a=qb     or     p'  =  q  -.        .     .     (12) 


pf  =p  tan2  0, 


PP 


(13) 


FIG.  38. 


FIG.  39. 


3.  The  same  analysis  may  be  applied  to  an  element 
at  the  inner  face,  Fig.  39,  where  <$>'  is  the  inclination  to  the 
vertical;  but,  for  the  reservoir  full,  the  intensity  of  water 
pressure,  horizontal  or  vertical,  at  c,  and  in  this  case 
represented  by  w,  must  be  taken  into  account. 


200  HIGH   MASONRY    DAM    DESIGN 

Under  these  circumstances, 

pb=qa  +  wb,     .     .     .     ,     .     .     (14) 
and 

p'a=qb  +  wa,     .     .     .     ...»     (15) 


(16) 
and 

p'=q  tan  <f>'  +  w  ......     (17) 


When,  as  is  usually  the  case,  the  vertical  component 
of  water  pressure  acting  along  the  back  is  neglected,  the 
above  equations  become, 

p  =  qcot<j>',     ......     (18) 

p'=q  tan  <f>'  +  w,   .....     (19) 

'y     ......     (20) 


and 

pf  =  tan2  (/>'  +  w  .......  (21) 


4.  If    an    element     at     the     down 

X 
/^v  stream  face  be  again  considered,  since 

tne    shear    on    the    outer   face    DC    is 


A^     q  —    zero,  that    on    a    plane    AD    perpen- 
\P  dicular  to  DC,  must  be  zero  also,  and 

hence  the   stress  d    on  A  D  is  wholly 
FIG'4°-  normal. 

The  total  pressure  on  AD  is  therefore, 

f-AD=f-bcos<f>.  (22) 


HIGH    MASONRY    DAM    DESIGN 


201 


The    vertical    component    of    this    is   /-6cos2<£,    because 


or, 


pb=f-bcos2  <f>, 


P 


p  sec2  <£, 


(23) 


(24) 


which  is  the  maximum  intensity  of  normal  stress  at  the 
outer  face. 


FIG.  41. 

5.  To  determine  the  planes  of  principal  stress,  i.e., 
planes  upon  which  the  stress  is  wholly  normal,  and  also 
the  intensity  of  that  stress,  we  may  assume  the  conditions 
indicated  in  the  figure. 

The  total  stress  on  c  then  is  jc ;  its  vertical  component 
jc  cos  6  =fb,  and  its  horizontal  component  fc  sin  6  =/a. 

When  2V  =  o  and 


,    .     .     (25) 


-p  =  qcot  0,         .     (26) 


202  HIGH    MASONRY    DAM    DESIGN 

The  difference  of  these  two  equations  gives, 


p-p'=q  (cot  0-  tan  0)  =q  .     (27) 


2  tan  0          20 


This  equation  gives  a  plane  upon  which  there  is  none 
but  normal  stress. 

To  determine/,  multiply  equation  (25)  by  (26). 

(f~P)(f-p')=q2,          ..'..-    (29) 
whence, 

q*)}-      .     .   -(30) 


This  will  give  two  values  of  /,  which  correspond  to  the 
two  principal  planes  of  stress,  the  stress  being  compressive 
when  /  is  positive,  and  tensile  when  /  is  negative.  There 
can  be  no  tension  when  pp'^.q2. 

Determination  of  the  vertical  unit  stress  at  any  point 
of  a  horizontal  plane  joint:  From  the  law  of  the  trapezoid, 
we  have  the  pressures  at  the  upstream  and  downstream 
toes  represented  respectively  as  follows: 


W,       .....     (31) 

jr 

W.  .     .     (32) 


The  resultant  is  supposed  to  act  within  the  middle 
third.  If  xf  represent  any  point  along  EB,  measured 
from  E,  then  /?,  the  pressure  at  x,  is  given  by, 

iv,         ..'.     .     .     (33) 


HIGH    MASONRY    DAM    DESIGN 
i 

while  the  total  noHnal  stress  from  E  to  xf  is,  by  integration, 


(34) 


To  find  the  unit  shear  on  vertical  or  horizontal  planes, 
we  have  but  to  consider  a  slice  of  dam  between  two  hori- 
zontal joints  one  foot  apart,  extending  from  the  inner  to 
the  outer  face,  a  distance  x  along  the  lower  joint.  (The 
back  is  supposed  to  slope  .02  feet  for  each  foot  in  height). 


"I"  a--Q-nr 

a?,! I 


FIG.  42. 


|p 
FIG.  43- 


The  vertical  forces  acting  are: 

(1)  A    uniformly    varying    stress    on    the    upper   joint 
acting  downward. 

(2)  The  same  on  the  lower  joint  acting  upward. 

(3)  The  weight  of  the  strip. 

(4)  The  shear  on  the  vertical  face  at  x. 
For  equilibiium, 


(35) 


Pf  and  P  may  be  obtained  as  indicated  in  the  previous 
demonstration. 


/ 

204  HIGH   MASONRY    DAM    DESIGN 

The  above  value  of  q\  is  the  average  unit  shear  at  the 
depth  taken,  but  a  similar  value  q^  may  be  determined 
at  a  depth  one  foot  below.  Under  these  circumstances 

— — —  is  the  average  of  the  two,  and  may  be  said  to  be 
approximately  equal  to  the  shear  at  the  depth  of  the  joint 
between  the  two  slices. 

Q' 


'/  . 

X—  0.01 

X                                              -H 

M 
G  •«  —  - 
N 

p. 

07+0.01 

FIG.  44. 

To  find  the  normal  unit  stress  on  a  vertical  plane,  a 
similar  section  to  that  just  used  may  be  employed;  but  the 
horizontal  components  are  now  to  be  equated  for  equilib- 
rium. 

Let  h  =  the  horizontal  water  pressure  at  the  assumed 

depth. 

Q'  =  total  shear  on  upper  face. 
0=  total  shear  on  lower  face. 
p'  =  average  normal  stress. 

q\  and  q^  =  the  intensities  of  horizontal  shear  at  the  points 
indicated. 

Q'  and  Q  may  be  found  by  integrating  q\  and  q^  be- 
tween the  proper  limits. 

•  '-'..-.    f-h  +  ff-Q.    .     ....     .     (36) 

This  value  of  pf  is  assumed  as  the  average  intensity 
on  the  vertical  plane  and  as  the  unit  intensity  on  the 


HIGH    MASONRY    DAM    DESIGN  205 

'> 

vertical  plane  at  "a  point  midway  between  the  two  hori- 
zontal planes. 

Three  general  formulae  may  be  written  for  p,  q,  and  pf 
which,  it  has  been  suggested,  be  put  in  the  following  form: 

p  =  a  +  bx, (37) 

(38) 
(39) 


\ 


I 

'Jfc 


APPENDIX  I 


DERIVATION  OF   CANTILEVER  EQUATIONS 

THE  expression  for  the  moment  of  inertia,  J,  of  the  hori- 
zontal cross-section  of  the  cantilever  contained  between 
two  vertical  radial  planes  of  the  arched  dam,  and  the 
two  fundamental  equations  for  deflection  represented  by 
A0  and  0n,  will  now  be  derived.  Th©  derivation  of  the 
expressions  for  these  last  two  quantities  for  the  special 
case  of  an  arched  dam  of  rectangular  vertical  cross-section 
will  also  be  indicated. 

Derivation  of  the  expression  for  the  moment  of  inertia,  I, 
of  the  horizontal  cross-section  of  the  cantilever  at  the 
distance  x,  below  the  given  origin,  or  Eq.  (i)  of  page  146: 


36Rn(2Rn-Bx) 


Referring  to  the  nomenclature  of  page  145,  and  to 
Fig.  27,  it  is  evident  that  l=Bx.  (Cf.  Fig.  45  and  Fig.  46.) 

If  we  assume  two  vertical  radial  planes  to  intersect 
the  dam,  i  foot  apart  at  the  up-stream  face,  the  plan  of 
the  section  to  be  considered  will  be  indicated  by  the  shaded 
area  in  Fig.  45.  It  will  be  sufficiently  exact,  however,  to 
substitute  the  chords  for  the  curves  themselves  as  limiting 
this  area.  Furthermore,  where  the  up-stream  face  of  the 

section  under  consideration  is  battered,  the  average  value, 

207 


HIGH    MASONRY    DAM   DESIGN 


between  the  two  successive  load  points,  may  be  assumed 
for  the  up-stream  radius,  /?,,  as  local  variation  in  this  respect 


— By  >-, 


FIG.  45. 

* 

has  been  treated  as  negligible  in  the  derivation  of  the 
general  expression  for  7. 

The  resulting  expression  will  then  be  transformed  by 
the  substitution  of  like  terms  into  one  corresponding  to 


CentroiJ 

!_«_+_* 


FIG.  46. 

the  nomenclature  of  Fig.  45,  which  wifl    be    the  desired 

Eq.  (i). 

The  reciprocal  of  Eq.  (i)  will  then  be  modified  by 
factoring  its  denominator,  in  order  to  render  it  integrable 
in  further  operations  where  it  may  occur. 


APPENDIX  I  209 

I 

The  general  expression  "for  the  moment  of  inertia,  /,  of 
Fig.  46,  is 

I=fXY*dY.       ......     (a) 

in  which 


Substituting  this  value  of  X  in  Eq.  (a)  and  designating 
the  limits  of  integration  as 

f/F.+aFV 

+ 

and 

V=         2' 
s 

there  results: 


/-\ I  — ^— : 
_/-T 


F-F,  r*5<Ka.^_ 


Performing  the  above  indicated  integrations, 


rtrpr'«»w 


_   _ 


F-F  'M 


Completing  the  above,  gives 


-3(F-F1)[(F1+2F) 


*-(2F1+F)*]\, 


210  HIGH    MASONRY    DAM    DESIGN 

which  may  be  reduced  to 


T  _  *-    \->-    1  /  J\ 

^      /     T^  -  7~*\  •*••••  \^*'/ 

To  transform  Expression  (d)  into  the  form  of  Eq.  (i), 
substitute  in  (d),  i  for  F,  i  —  —  for  Fi,  and  5^  for  /, 
as  indicated  in  Fig.  45 .  There  will  then  result 


j 


36Rn(2Rn-Bx) 
which  is  Eq.  (i)  of  page  207.^ 


To  render  -=  integrable  by  factoring. 


From  the  above  equation  we  have 


By  factoring  ^2~~D~^+6(^)    m  tne  denominator  of 

(e)y  into  the  typical  factors,    (x+d)(x+b),   there  results, 
after  reduction,  the  following  integrable  form : 


APPENDIX   I  211 

ill 

Development  of  the  equations  for  Deflection  in  the  Cantilever 
for  EAa  and  E$n: 

Eqs.  (9)  and  (10)  of  page  150  are  the  fundamental 
differential  expressions  from  which  are  derived  Eqs.  (13) 
and  (14),  for  the  cantilever. 

As  there  stated,  the  expression  for  I  and  that  for 
M  are  employed  by  substituting  them  in  Eqs.  (9)  and 
(10)  and  integrating.  These  operations  are  indicated 
below. 

V 

Derivation  of  E£n. 
Eq.  (9)  of  page  150  is 


Eq.  (n)  of  page  150  is 


Eq.  (/)  of  p.  210,  Appendix,  may  be  written 


The  general  integral  of  Eq.  (9)  may  be  written 

(g) 


Substituting  the  expressions  for  Mn  and  j  in  Eq.  (g)  and 
reducing,  gives 


212 


HIGH    MASONRY    DAM    DESIGN 


-gyv 


tdx 


»     *  -  (3  -  V3) 


+(Pidi+P2d2+ 


*2[*  -  (3  +  V?)§']  [*  ~  (3 


*[*  -  (3  +  V3)§]  [x  -  (3  -  V3)§n] 


+Pn 


.To  integrate  Eq.  (h)  with  respect  to  #,  between  the 
limits  c^n+1  and  c^ra,  it  will  be  necessary,  first,  to  determine 
the  integrals  for  the  four  fractions  containing  dx  in  the 
numerator  of  each,  and  the  denominators  as  written  above, 
functions  of  descending  powers  of  x,  beginning  with  x3, 
as  a  factor  for  the  first  denominator. 

These  separate  integrations  may  be  accomplished  by 
expanding  each  fraction  into  a  series  of  partial  fractions 
by  the  method  of  undetermined  coefficients  and  then 
integrating  each  term  of  the  series.  This  will  result  in 
Eqs.  (f),  (/),  (k),  and  (/).  These  equations  will  serve  for 
the  derivation  of  E0n,  as  well  as  for  EAn. 

According  to  the  theorem  of  undetermined  coefficients, 
there  may  be  written  for  the  first  fraction  of  Eq.  (h) 
expanding  in  ascending  powers  of  x  and  distinguishing 


213 

B,  the  coefficient,  b^  a  vertical  letter  from  B,  the  batter, 
an  inclined  letter, 

dx  _Adx  Jbdx     Cdx 

1 5"     I o~ 

X  X2          X3 


*-(3  +  V3)^||*-(3- 

Edx 





Clearing  of  fractions,  collecting  the  terms  in  the  second 
member  involving  like  powers  of  x,  equating  coefficients 
of  like  powers  of  x  in  the  two  members  and  solving,  gives 


»- 


The  integral  of  this  first  fraction  may  thence  be  written 

doc 


S_IB\*  r  °+idx  i/B 

3«W  .  *      6 


-MrT 


B 


214  HIGH    MASONRY    DAM    DESIGN 

The  definite  integral  of  which  is 

r  d»+i  dx 


[/-  R  1  /—R 

*- (3  +  vfrg  \\x  -  (3  -  v  3)— n 
JD  J  L  n 

$B^          d         B^(d      d  ^      R2f/72 

"1  s   T~t    A    -^Oe          J  I  f    T~>    1  _7      J 


The  second  fraction  of  Eq.  (/&),  by  the  method  of  unde- 
termined coefficients,  may  be  expanded,  thus: 

dx  Adx     Edx 

"1         " 


?  z? 

whence  there  follow  the  values : 


36 

3U  ' 


APPENDIX  I  215 

4* 
The   integral    of    the    second   fraction    may    then    be 

written : 

r  dn+i dx 

/        r  R  i  r  R 

Jdn      y?\x - (3 -f-V^)— w 1 1 x - (3  - v^)-^ 

r*  ^n  +1  f*  ^"n  -4- 1 

.  I(J/^    ^+l\Rj  J      J 


,      -          n+l 


36 


The  definite  integral  of  which  is 


,       dn+1 
dn 


i-(3+VJ)f 


216  HIGH    MASONRY    DAM    DESIGN 

Similarly,  for  the  third  fraction  of  Eq.  (h), 
dx  Adx 


Edx Cdx 


Proceeding  as  before,  there  result, 


12 


The  integral  of  this  third  fraction  follows 
dx 


rdn+i 

J*.          X\X- 


- 


VJ-i/gy  f 
»     \RjJda 


12 


The  definite  integral  of  which  is: 


APPENDIX  I  217 

Df 

dx 


-  +         log          +         -l        log 

&  U&e 


I27? 


And  for  the  fourth  fraction  of  Eq.  (h), 

dx 


W 


Adx  Edx 

= —  H 


From  which : 


L 


The  integral  of  this  fourth  fraction  is 


B 


218  HIGH    MASONRY    DAM    DESIGN 

The  definite  integral  of  which  is : 
doc 


L 


The  integration  of  Eq.  (/&)  may  now  readily  be  accom- 
plished by  simply  substituting  in  that  equation  the  right- 
hand  members  of  Eqs.  (i),  (/),  (k),  and  (/),  above,  for 
their  corresponding  left-hand  members,  as  they  occur  in 
Eq.  (fe). 

By  performing  the  indicated  operations,  collecting 
terms,  and  writing,  in  place  of  the  Napierian  logarithm, 
the  conversion  factor,  2.30259,  so  that  common  logarithms 
may  be  used  in  computations,  there  results  Eq.  (13), 
for  EAn,  as  finally  written  on  page  153.  (Note  that  ln 
has  been  substituted  for  Bdn  and  ln+1  for  Bdn+1,  in  this 
expression.) 

Derivation  of  E0n. 

Eq.  (10),  of  page  150,  is: 

,      Mdx  x 


The  general  integral  of  which  may  be  written  : 

E  CdB  =  Cjdoc.  .     .     .     .     .     .     (m} 


APPENDIX    I  219 

i 


Substituting  .(/),  -  from  page  211  and  the  expression 

for  Mn,  Eq.  (n),  in  the  above  Eq.  (m),  and  collecting  terms, 
gives : 


(Pl  +P 


[*  -  (3  +  V3)|"  ]  [*  -  (3  -  VJ)|»] 


*  -  (3  +  VI)|"]  [*  -  C3  -  vD§ 


Obviously,  the  definite  integrals  of  Eqs.  (i),  (/),  and 
(k),  as  developed,  are  directly  applicable  to  Eq.  (n),  above. 

Substituting  the  right-hand  members  of  Eqs.  (i),  (/), 
and  (£),  for  their  corresponding  left-hand  members,  as 
they  occur,  in  Eq.  (n),  performing  the  indicated  operations, 
collecting  terms,  together  with  substitution  of  the  con- 
version factor,  2.30259,  and  ln+l  and  /„,  as  explained  in 
the  treatment  of  Eq.  (h),  above,  will  result  in  Eq.  (14), 
for  Edn,  as  finally  written  on  page  153. 

Deflection  equations,  for  Arched  Dam  of  rectangular,  vertical 
cross-section. 

Derivation  of  ElAn. 

The  moment  of  inertia,  being  a  constant,  becomes  a 
factor  with  E  on  the  left  side  of  the  equation. 


220  HIGH    MASONRY    DAM    DESIGN 

As  before,  Eq.  (9)  applies,  or,  as  dn  =  na, 


Whence 


X(w  +  l)a  f*(n  +  l)a 

dAn=  I  M(x-nd)dx.      t  - .     (p) 

i  Jna 

Substituting  the  value  for  Mn  from  Eq.  (15)  of  page  167 
and  performing  the  integrations  indicated  and  collecting 
terms,  Eq.  (16)  results  directly. 

Derivation  of  EIOn. 

Eq.  (17)  of  page  167  may  be  similarly  derived  by  means 
of  Eq.  (10),  which  may  be  written 

El  C    l}°den  =  (         Mdx, 


X(»+l)a  /~(n+l)a 

dBn  =  j 
I  JtM 


into  which  the  value  of  Mn,  from  Eq.  (15),  is  substituted 
and  integrations  performed,  terms  collected,  etc.,  as  before. 


APPENDIX   II 


MOVEMENTS  AND  STRESSES  IN  AN  ARCH   SUBJECTED  TO 
A   UNIFORM,  RADIAL  LOAD 

With  Derivation  of  Eq.  (8},  of  page  149,  for  Arch  Crown  Deflection. 

The  nomenclature  of  page  145,  together  with  designa- 
tions of  Fig.  47  and  such  other  as  may  be  immediately  per- 
tinent, will  obtain  in  the  following  discussion  which  is 
largely  adapted  from  a  discussion  by  the  late  R.  Shirreffs.* 

In  Fig.  47,  line  1-2 '-3'  represents  one-half  of  the  axis 
of  a  segmental  arch  ring  in  its  unloaded  position.  For 
convenience  of  reference  the  line  $f-0'  may  be  assumed  as 
vertical  and  passing  through  the  crown  of  the  arch.  The 
abutment,  or  skewback  supporting  the  arch,  may  be  assumed 
to  be  in  the  line  1-4-0'.  <£„,  then,  is  one-half  the  central 
angle  of  the  arch  span,  and  rn  the  radius  of  the  axis. 

In  elucidating  the  various  analytic  expressions  for 
effects  of  loading  both  as  to  stressing  and  deflecting  the 
arch,  certain  changes  in  position  are  imagined. 

These  are,  in  order,  as  follows : 

Beginning  with  the  unloaded  arch,  one-half  of  which  is 
shown  in  Fig.  47,  and  assuming  it  either  to  be  a  portion  of 
a  closed  ring  or  to  rest  upon  frictionless  abutments,  a  radial 
loading  of  intensity  qn,  will  produce  a  shortening  of  the 

*  Trans.  Am.  Soc.  C.E.,  Vol.  LIII,  p.  163. 

221 


222 


HIGH    MASONRY    DAM    DESIGN 


original  length  and  a  reduction  of  the  radius  length  by  a 
proportionate  amount,  /.  The  half  arch  will  be  moved 
into  a  new  position  4-5-6.  It  will  be  in  equilibrium  under 
only  the  axial  thrust  qnrn. 

Secondly,  assume  the  arch  severed  at  the  crown  and 
the  right  half,  removed,  replaced  by  a  crown  thrust  =  qnrn 


FIG.  47. 

and  the  remaining  half  arch  moved  outward  until  Point  4 
again  coincides  with  Point  i.  The  perpendicularity  of  the 
axis  at  the  skewback  remains  unchanged. 

The  center  of  the  arch  will  be  at  o  instead  of  0' ;  Point  6 
will  be  in  the  position  of  Point  3,  at  the  horizontal  distance 
k'  from  the  vertical  through  the  middle  of  the  span  or 
crown  of  the  arch,  and  at  the  vertical  distance  Ac  below 


APPENDIX   II  223 

I 

the  original  crown  -ef  the  arch.     The  crown  joint  will  still 
be  vertical. 

Thirdly,  in  order  to  restore  the  integrity  of  the  arch, 
under  its  loading,  the  crown  thrust,  qnrn,  must  be  so  di- 
minished that  under  the  combined  action  of  this  diminished 
thrust  and  the  loads  on  the  half  arch,  the  curved  beam 
1-2-3,  now  considered  fixed  at  the  abutment,  shall  be 
deflected  through  the  horizontal  distance  k'  and  as  the 
original  crown  thrust,  qnrn,  just  holds  the  arch  in  equilibrium 
against  the  action  of  the  loads,  a  force  Hf  applied  at  the 
crown  and  equal  to  the  necessary  diminution  of  qnrn,  H' 
acting  therefore  to  the  right,  will  cause  a  movement  iden- 
tical with  that  through  k' '. 

Fourthly,  the  crown  joint,  which  will  have  been  de- 
flected through  an  angle  0,  by  this  movement,  must  again 
be  made  vertical  in  its  new  position.  This  can  be  accom- 
plished only  by  the  application  of  a  moment,  Me. 

Fifthly,  the  total  movement  of  the  arch  at  any  point 
will  be  obtained  by  combining  the  movement  resulting  from 
axial  stress  with  those  movements  produced  by  the  force 
H'  and  the  moment  Mc. 

The  above  considerations  will  next  be  treated  analy- 
tically. 

Derivation  of  expression  Ac,  under  axial  thrust. 

Assume  the  arch  ring  to  be  of  thickness  =  /„  and  a  depth 
(normal  to  the  plane  of  the  paper  in  Fig.  47)  of  i. 

Let  X   =  shortening  of  the  half  length  of  arch  shown  in 

Fig.  47. 

L  =  curved  length  of  arch  from  abutment  to  crown. 
/„  =  area  of  radial  vertical  cross-section  of  arch  ring. 
E  =  modulus  of  elasticity. 


224  HIGH    MASONRY    DAM    DESIGN: 

Then 

^--  =  intensity  of  stress  in  ring.    .     .     .     (i) 


-=  strain  due  to  stress.   .  .....     (2) 

j  --/ 


Whence  X=      n '    .     .     .     .     (3) 

Since  the  ratio  of  the  shortening  of  the  arch  to  its 
original  length  is  equal  to  the  ratio  of  the  change  in  length 
/',  of  the  radius  to  its  original  length, 

L=7a 
Whence 

/=^f •     (4) 

Combining  Eqs.  (3)  and  (4)  gives 


And  as  kr  =/  sin  $n  (see  Fig.  47), 

y-sffl*;     ....     (5) 


and  as  Ac=/(i  -cos  0n),  from  Eq.  (40),  there  follows,  in 
this  connection: 


(6) 


ABPfeNDIX   II  225 

> 

At  a  distance  of  </>  degrees  from  the  crown  there  will  obtain 

A^  :  Ac  =  fa  —  <£  :  <t>n , 
therefore 


Derivation  of  expression  for  Hf,  or  diminution  of  stress 
qnrn. 

The  slight  reduction  in  the  compression  of  the  arch  ring 
due  to  diminishing  qnrn  by  the  amount  Hf  is  neglected  in 
the  following. 

The  general  expression  for  the  differential  deflection 
ds,  of  a  beam  is 

»      Mxdx 


in  which  x  is  referred  to  any  point  in  the  beam's  axis, 
I  ds  is  the  deflection  with  respect  to  that   point  and  M 
is  the  bending  moment  about  that  point.     (See  Point  2  of 
Fig.  47-) 

In  this  case 

M=H'rn(i  —cos  <£), 

x  =  2rn  sin  — ,  sufficiently  close. 
2 

dx  =  rndcj>, 

J  3 
I--2k 

12 

Substituting  these  last  expressions  in  the  general  expres- 
sion for  ds,  results  in 

ds'  =       * — (i  —cos  0)  sin  -d^>. 

2 


226  HIGH    MASONRY   DAM    DESIGN 

But  the  horizontal  component  of  ds'  is  dk', 


since  (Fig.  47)  ds',  approaching  zero,  sensibly  coincides  in 
direction  with  the  line  3-7,  the  tangent  to  the  arc  ds'. 
Therefore 

*dt.     ....     (7) 


Integrating  Eq.  (7)  between  the  limits  <f>n  and  o  and 
equating  the  result  with  the  right-hand  member  of  Eq.  (5), 
for  k',  gives 

2  Sin  * 


. 
12  3  0n  +  sin  4>n  cos  0re  -  4  sin 


(8) 


Derivation  of  expression  for  moment  Mc,  or  the  moment, 
the  effect  of  which  is  to  render  the  crown  joint  vertical, 
in  position. 

The  moment  Mc  must  cause  the  same  angular  deflection, 
j8,  in  the  entire  beam  as  the  force  Hf. 

The  general  equation  for  angular  deflection  is 


x 

x  and  M  are  again  taken  with  reference  to  the  same  point, 
which  may  be  any  point  at  <£,  rn  in  beam.     (See  Point  2.) 
From  the  foregoing  considerations,  substitute  for   . 

M  =MC  =H'rn(i  -cos  0), 


12 


APPENDIX   II  227 

i 

in  the  above  expression  for  $0,  whence 

dp=— ^^-(i-cos 


Integrating  both  expressions  for  dp  between  the  limits 
of  <f>  =  4>n  and  0  =  o  and  equating  the  results  give : 

M.=H'r  *•-*** +*.  (9) 

<t>n 

Substituting  the  value  of  H'rn  from  Eq.  (8)  above  in 
Eq.  (9)  gives  for  Mc, 

M  =gn42/<An-sin  <j>n\ 2  sin  0M /     , 

12  \        <t>n       /30n+sin  <f>n  cos  (j)n  -4  sin  <J>n' 

Mc  is  opposite  in  effect  as  to  resulting  deflection,  to  that 
produced  by  Hf. 

The  deflection  of  the  arch  under  the  combined  action  of 
Hf  and  Mc. 

The  object  of  this  phase  of  the  analysis  is  to  determine 
the  combined  movement  in  a  radial  direction,  due  to  H' 
and  Mc,  for  this  radial  deflection  is  the  arch  deflection 
which,  in  a  curved  dam  will  produce  stresses  in  the  vertical, 
or  cantilever  beams. 

The  general  expression  for  the  deflection  of  a  beam 
again  serves,  i.e., 

ds=Mxm- 

The  reference  point  or  origin  now  will  be  some  other 
point  than  2,  of  Fig.  47,  such  as  #,  at  any  angle  co  with  the 
radius  through  Point  2. 


228  HIGH   MASONRY   DAM   DESIGN 

The  two  moments  considered  with  respect  to  Point  x 

will  be 

H'rn[i  -cos  O  +  co)]  and  Mc. 

The  resulting  moment,  etc.,  will  therefore  be 
M=#V«[i-cos  (0  +  co)]-Mc. 

CO 

X  =  2Tn  Sin  —  . 
2 

I  3 

dx=rndu     and    J=—  . 
12 

Substituting  these  in  the  general  expression  for  ds  above 
gives 

'*  ={H'rn[i  -cos  (*  +  «)]  -Mc]  sin  -du. 

2 


Let  the  radial  component  of  this  movement  ds'^  be 
represented  by  dD'+. 
Then 


therefore 

dD=\H'rnV  -cos  (0  +  co)]  -Me!  sin 


Integrating  this  last  expression  for  dD'0,  between  the 
limits  co  =  (<£„  —  4>)=7  and  co=o  and  substituting  J/Vn  and 
Mc,  as  written  in  Eqs.  (8)  and  (10)  respectively,  gives 
for  D', 


in  which  C  has  the  general  value  given  in  Eq.  (na),  or 


APPENDIX   II  229 

~     \2  sin  0n/  •"    v   .  sin  $,          .        N 

G  =   •    (i  —  cos  7)  H (27  —sin  27) 

L      4>n  2 

COS  <p   /  \   I         /     ' 

-   (COS  27  —  l)      -r-  (  -r 


When  0=o  and  therefore  7  =  </>„,  the  value  of  C  is  as 
given  in  Eq.  (u&),  or 

*?(i  -cos  0n)  +Kcos  20n-i) 


sin  0n 


and  for  the  semicircle,  since  <£n  =-, 

2 


(nc) 


To  combine  the  effect  of  the  axial  thrust  or  Ac  with  the 
crown  deflection  as  given  in  Eq.  (116)  and  Eq.  (n),  there 
should  be  noted  the  following : 

The  last  term  in  the  numerator  of  Eq.  (nfr)  involves 
2  0».  This  may  be  written  in  terms  of  <£„,  for 

J(cos  20rt-'i)  =(cos20n-i), 

or2 

also,  the  coefficient  of  C  in  Eq.  (n),  -^f-  is  the  same  as 

hln 

the  coefficient  of  (i  —cos  0n)  in  Eq.  (6)  for  Ac.  Hence  the 
combined  trigonometric  function  of  <£n  designated  as  CCC, 
Fig.  28,  which  is  the  coefficient  plotted  as  a  curve,  referred 
to  before,  for  the  common  factor  of  Eqs.  (n  )  and  (6), 


230  HIGH    MASONRY    DAM    DESIGN 

or2 

viz.,   *£?-•     This   combination  is  written   as  Eq.    (8)    on 
Eln 

page  149  for  De. 

Derivation  of  the  moment  M  $,  at  any  point,  <j>  degrees 
from  the  crown. 

The  moment  is  HVn(i  -cos  0)  -Mc. 

Substituting  the  values  of  H'rn  and  Mc  as  written  in 
Eqs.  (8)  and  (10),  and  reducing  give 


(12) 


sin 


M0  becomes  o,  or  there  is  a  point  of  contra  flexure  in  the 
curved  beam  used  as  an  arch  under  a  uniform  radial  load, 

when 

sin  0n 
cos  <£  = ™, 

or  in  the  semicircle  when 

cos  0=-, 

7T 

or  at  about  50°  from  the  crown. 

The  stresses  of  tension  or  compression  produced  by  M^ 
at  any  point  in  the  arch  may  be  combined  with  the  axial 
stress  qnrn,  whence  the  resultant  stress  at  that  point  may 
be  obtained.  The  force  H't  except  in  very  flat  arches,  may 
usually  be  neglected  in  getting  the  resultant  stresses. 


APPENDIX   III 

CROSS-SECTIONS   OF   EXISTING  MASONRY   DAMS 

THIS  selection  of  cross-sections  of  notable  masonry 
dams,  following  Table  XIV,  is  arranged  in  chronological 
order  to  illustrate  the  evolution  of  the  masonry  dam, 
the  design  of  which  has  been  largely  a  matter  of  following 
precedent,  in  many  cases. 

The  series  begins  with  a  few  of  the  heavy,  Spanish 
type  and  continues  through  the  French  designs,  such  as 
those  of  de  Sazilly  and  Delocre,  to  the  types  of  present-day 
construction,  as  illustrated  by  foreign  and  American 
(United  States)  examples. 

With  respect  to  the  design  of  de  Sazilly,  reference  to 
whom  was  made  early  in  Chapter  I,  it  should  be  stated 
that  his  analysis  resulted  in  a  cross-section  that  was 
necessarily  stepped,  and  Delocre  suggested  substituting 
curved  faces  in  place  of  the  steps,  thereby  effecting  an 
appreciable  saving  of  material  in  construction. 

Attention  is  directed  to  the  fact  that  all  of  the  sixty- 
one  sections  could  not  be  shown  to  the  same  scale;  but 
each  cross-section  is  sufficiently  dimensioned  so  that  com- 
parisons are  possible. 

The  various  methods  of  designating  batters  are  also 
apparent. 

The  later,  overfall  types  only,  have  been  grouped 
separately. 

231 


232  HIGH    MASONRY   DAM   DESIGN 

Table  XIV  contains  a  page  index  of  the  cross-sections, 
together  with  other  data  concerning  each,  for  convenience 
of  reference. 

In  the  table,  under  "  Type  of  Section," 

G=  gravity  section  (depending  on  gravity); 
A  =  arch  section  (depending  on  arch  action) ; 

and  under  "  Type  of  Service," 

N.O.  =  non-overflow; 
S.  =  spill  way  dam. 

A  dash,  in  the  column,  headed  "  Radius,  etc.,"  denotes 
a  straight  alignment  of  dam. 


APPENDIX    III 


233 


>      S 

t— I         «jj 

X  Q 

9  I 

W    o 

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ign 
or 
lder 


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Bu 


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234 


HIGH    MASONRY    DAM    DESIGN 


Remarks. 

N.  Y.  City  Water-supply.  Cyclop, 
masonry;  concrete  face  blocks. 
Chemnitz  Water  Supply.  Rubble 

masonry. 
U.  S.  Reclamation  Service. 
Remscheid  Water  Supply. 
Near  Pittsfield.  Expansion  joints 
similar  to  those  of  Kensico  Dam. 
Near  St.  Etienne.  Delocre  de- 
sign. Is  watertight  dam. 
Rubble  and  cut  stone. 
Partially  failed  in  1885.  Rubble. 
Foundation,  soft  rock.  Buttressed, 
and  in  1896  earth  filled  on  face. 

Poilo^in  rRRr  flnnrl  ti  ft.  nvpr  rrfist. 

Repaired  in  1883-7.  New  profile 
not  found. 
Failed.  Rubble  masonry. 
Almeria.  Two  dams.  Rubble  with 
cut  stone  facing. 
At  foot  of  Indian  Lake,  N.  Y. 
Thermophones  installed  to  detect 
internal  temperature  changes. 
Tar  and  asphalt  waterproofing  in 

up-stream  face.  Drainage  system. 
Rubble.  Has  had  15  ft.  depth  of 

water  on  crest. 
For  Denver  Water  Supply.  Gran- 
ite in  Portland  cement  mortar. 

Cyclopean  masonry. 
Straight  for  half  its  lehgth.  Rest 
arched. 
Three-coat,  pitch  waterproofing  in 

up-stream  race. 
New  York  City  Supply.  Spillway 
at  one  end. 

•      '$ 

03 

Location. 

New  York  . 
Germany.  . 

£  £  w      G     'Sb  .2  G     .! 

£  o  ^    £    m  <i£    < 

i         .j|.S      £  £      a 

I 

U 

o    s£ 

! 

Continued 

Designers 
or 
Builders. 

American  .  .  . 
German.  .  .  . 

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APPENDIX    III 


235 


N.Y.City  Supply.  (Contract  Draw- 
ing.) Built  with  block  parapet. 
North  Platte  Reclam.  Project,  U.  S. 
Hydraulic  lime  concrete.  Rubble 
facing. 

S*           r«              •    (U                   **K              •           jj              •       •          «4_|       •                   ^                                            •            S^  *>            S.      • 

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*  Polygonal  in  Plan.  t  Under  construction,  1916.  J  300  ±  to  deeoest  foundation. 

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Olive  Bridge.  .  . 

Pathfinder  
Periyar  

^ii!  si  J  ill 

O3       o       pj            <D  ji       o       ft       3^        rt            <u  &      .tni-.cJr^>>K";^ 
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ca 

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236 


HIGH    MASONRY   DAM    DESIGN 


CROSS-SECTIONS   OF   EXISTING  DAMS 

(Scales  are  as  indicated  by  dimensions) 


61.30 


ALMANZA 

Spain 
Constr.  prior  to  1586 


T 


\ 


VAL  DE  INFIERNO 

Spain 
Constr.  1785-1791 


APPENDIX    III 


237 


Radius  =  Infinity- 
Plan  Pdlyyonal 


16.69 

f 

30.39 

44.08 


153.50 


115.20' 
PUENTES 


Guadalentin  River,  Spain 
Constr.  1785-1791.     Failed  1802 


GROS  Bois 

River  Brenne,  France 
Constr.  1830-1838 


238 


HIGH    MASONRY   DAM    DESIGN 


Radius=158  ft. 


ZOLA 

Nea   Aix,  France 
Constr.,  1843 


-Rad.=Inftnity- 


69.00— 


J 


SETTONS 

Yonne  River,  France 
Constr.  1855-1858 


APPENDIX   III 


239 


Radius=1312.4 


118.11 


TERNAY 

Ternay  River,  France 
(Bouvier  design) 

Constr.  1861-1867 


161.03 


FURENS 

Furens  River,  France 
(Delocre  design) 

Constr.  1862-1866 


240 


HIGH    MASONRY    DAM    DESIGN 


-Radius =Infiniej> 


n0.23'/c-19.95-'-j 88,39-— 

116.79'^ ! 


Profile  of  new 
part  of  Habra  dam 
differs  from  old,  but 
new  profile  has  not 
been  found. 


HABRA 

Habra  River,  A  Igiers 
(Old  profile) 

Constr.  1865-1871 

Failed  1881 
Repaired  1883-1887 


U Radius= 


1325' 


151.91 


=. 126,96- 

BAN 

Ban  River,  France 
Constr.  1866-1870 


APPENDIX   III 


241 


BOYDS  CORNERS 

West  Branch  of  Croton  River,  New  York,  U.  S.  A. 
Constr.  1866-1873 


U — Radius= 


im* 


27.88'     /- 63,96- 


GlLEPPE 

Belgium 
Constr.  1870-1875 


242 


HIGH    MASONRY   DAM    DESIGN 


•Radius=4iO' 


84.29' 


162.08 


-451.59— 


VlLLAR 

Lozoya  River,  Spain 
Constr.  1870-1878 


nterforts  are  IG.l'widt 
and  about  55 'centres. 


PONT 

ArmaQon  River,  France 
Constr.  1878-1881 


APPRNDIX    III 


243 


'Radius.^  Infinity 


1.97. 


13.12' 

29.52' 

50.0C' 
64.03' 


BOUZEY 

Aviere  River,  France 
Constr.  1879-1880.     Overturned,  1895 


BOUZEY 

Large  scale  profile  showing  reinforcement  of  1888-1889 
Overturned  in  1895,  580  feet  going  out 


244 


HIGH    MASONRY    DAM    DESIGN 

.  Radius  =  Infinity 


59.05 


92.22' 


114.83 


02.69!' 


HAM  iz 

Algiers 
Constr.  1880-1885 


U Radius  =  210' 


HlJAR 

Two  such  dams  across  the  Martin  River,  Spain 
Constructed  j  ^ 


APPENDIX    III 


245 


187.8 


134.52 ' 

GRAN  CHEURFAS 
Sig  River,  Algiers 
Constr.  1882-1884 


Radius  =  Infinity 


66.56' 
MOUCHE 

Mouche  River,  France 
Constr.  1885-1890 

Seven  temperature  cracks  appeared  during  winter  following  completion,  and 
in  the  summer  temperature  change  bowed  the  structure  slightly. 


246 


HIGH    MASONRY   DAM    DESIGN 


BRIDGEPORT 

Mill  River,  Connecticut,  U.  S.  A. 
Constr.  1886-1887 


•Radius  =-213.3' 


SWEETWATER 

Sweetwater  River,  California,  U.  S.  A, 

(Schuyler  design) 

Constr.  1886-1888 
Additions,  1895 


AP?EKDIX    III 


247 


105' 


-110- 

BEETALOO 

Australia 

Constr.  1886-1889 


-Rad.=  Infinity 
=160* 


99.8' 

TANSA 

Tansa  River,  India 
Constr.  1886-1892 


248 


HIGH    MASONRY    DAM    DESIGN 


THIRLMERE 

England 
Constr.  1886-1893 


262 


SAN  MATED 
California,  U.  S.  A. 
Constr.  1887-1888 


APPENDIX    III 


249 


159.93 

CHARTRAIN  OR  TACHE 
Tdche  River,  France 
Constr.  1888-1892 


(^Radius  =  .Infinity 


88.75' 


SODOM 
East  Branch  of  Croton  River,  New  York,  U.S.A. 

Constr.  1888-1892 


250 


HIGH   MASONRY   DAM   DESIGN 


16.95 


ESCHBACH 

Germany 
Constr.  1889-1891 


80.3^ 

LAUCHENSEE 

Germany 
Constr.  1889-1895 


APPENDIX    III 
i 


251 


PERIYAR 

Periyar  River,  India 
Constr.  1889-1896 


"jg    {4—  iRadius  -isio.oo' 


ElNSIEDEL 

Near  Chemnitz,  Germany 
Constr.  1890-1894 


252 


HIGH    MASONRY   DAM    DESIGN 


Rad.=  Infinity 


iR.=  Infinity 


Tmcus 

Titicus  River,  New  York,  U.  S.  A, 
Constr.  1890-1895 


NEW  CROTON 

(As  built) 

Croton  River,  New  York,  U.  S.  A, 
Constr.  1892-1906 


APPENDIX    III 


253 


[*-R.=  Inggitr 


,U-25.-75->! 
EL  416 1  _      _l 


El.  300'       Finished  

Surface  of  Earth//  81.49' 


Built  under  over- 
nging  cliff 


WACHUSETT 

Nashua  River,  Massachusetts,  U.  S.  A. 
Constr.  1896-1905 


•  Rad.=  Infinity 


INDIAN  RIVER 

New  York,  U.  S.  A. 

Constr.  1898 


254 


HIGH    MASONRY   DAM    DESIGN 


uilt  of  Granite  Rubble 


K- 80.-36- ->i 

i  t 

ASSUAN 

Nile  River,  Egypt 
Constr.  1898-1903.     Raised  in  1909  to  131',  by  increasing  entire  cross-section. 


BAROSSA 

Gawlor,  South  Australia 
Constr.  1899-1903 


APPENDIX   III 


255 


LAKE  CHEESMAN 
Colorado,  U.  S.  A. 
Constr.  1900-1904 


KOMOTAU 

Upper  Franconia, 
A  ustria 

Constr.  1900-1903 


256 


HIGH    MASONRY    DAM    DESIGN 
I 

[•« Radius  =  Infinity- 


\  Concrete 


SPIER'S  FALLS 

New  York,  U.S.A. 

Constr.  1900-1905 


MERCEDES 

Durango,  Mexico 
Constr.  1902-1905 


80.38 ' 


APPENDIX    III 


257 

URFT  RIVER,  NEAR 
AACHEN 

Germany 

''Prof.  Otto  Intze  de- 
sign) 

Constr.  1901-1904 


Salt  River,  Arizona, 
U.  S.A. 

Constr.  1905-1911 


258 


HIGH    MASONRY    DAM    DESIGN 


..  337.S97 


3^&ELe-v^£P-°^ 


K. 
-114rlO- - 

CROSS  RIVER 

Cross  River,  New  York,  U.  S.  A. 
Constr.  1905-1909 


SHOSHONE 
Wyoming,  U.  S.  A. 
Constr.  1905-1910 


APPENDIX    III 


259 


PATHFINDER 

North  Platte  River, 
Wyoming,JJ.S.A. 

Constr.  1906-1910 


CROTON  FALLS 
New  York,  U.S.A. 
Constr.  1906-1911 


7/t 

_¥         Y  EU55' 


'--' 


260 


HIGH    MASONRY   DAM    DESIGN 


OLIVE  BRIDGE  DAM,  ASHOKAN  RESERVOIR 

New  York,  U.S.A. 

Const.  1907-1914 

(See  page  96) 


KENSICO 

New  York,  U.S.A. 
Constr.  began  1912 


•d^MElev.3215' 


APPENDIX   III 
* 


261 


.Radius=  661.74 


ARROWROCK 

Boise  River,  Idaho,  U.  S.  A. 
Constr.  1912-1915 


262 


HIGH    MASONRY   DAM   DESIGN 


El.  264'         I18"3"!      Reservoir-full 


1 


-  -212,58 


ELEPHANT  BUTTE 

Rio  Grande,  New  Mexico,  U.  S  A. 

Constr.  began  1912 


APPENDIX   III 


263 


5'0' 


Selected  Materi'a 
.rolled  in  4"layers 


Note:-Top  of  dam  to  / 
slope  )^"per  ft.  from 
center  to  outsiue 
edge  of  ro'ping 


^ 

Granolithic  surface 
Overflow  E1.1585'j 


TYPICAL  SECTIO'N  OF  DAM 


'Expansion- joints  may  be  required  to  be 
built  with  or  without  moulded  blocks 


Steps  not  shown 


Class  A 


r 


HORIZONTAL  SECTION  AT 
EXPANSION-JOINT  AT  EL1542 

FARNHAM 

Mill  Brook,  Massachusetts,  U.  S.  A. 
Completed  1912 

Designed  for  ice- thrust  of  10,000  pounds  per  linear  foot  of  dam  length 
and  uplift,  intensity  due  to  full  head  at  up-stream  edge  of  joints  diminish- 
ing uniformly  to  zero  at  three-quarters  the  distance  to  the  down-stream 
edge  of  joints. 


264 


HIGH    MASONRY   DAM    DESIGN 
SPILLWAY  DAMS 


Radius  •=  Infinity 


Vyrnivy  River,  Wales 
Constr.  1881-1888 


Reservoir  Sid 


16.40' 11 


65.60 


BETWA 

River  Betwa,  India 
Constr.  about  1888 


APPENDIX    III 


265 


,(«— Radius=300  fU 


127.5 


LA  GRANGE  OR  TURLOCK 

Tuolumne  River,  Southern  California 
Constr.  1890-1893 


COLORADO 

Texas,  U.  S.  A. 

Constr.  1891-1892 


266 


HIGH    MASONRY   DAM    DESIGN 


SUDBURY 

Stoney  Brook  Branch 
of  Sudbury  River, 
Massachusetts, 
U.  S.  A. 

Constr.  1894-1897 


E1.900' 


BIG  BEND 

Feather  River,  Cali- 
fornia, U.  S.  A. 

Constr.   began     in 
1909 


APPENDIX    III 


267 


ASHOKAN  WASTE  WEIR 
New  York,  U.S.A. 
Constr.  1910-1912 
(See  page  104)  " 


El.  1595 
El.  1585', 

17-l<i  bare,  18" c.  to  c.,  16'o"long 
16-1)4 "bars,  18"c.  to  c.,  25'o"long 
W)-l)4"bars,  18"c,  to  c.,  33'8"long 


gelected  Material  \orr 
rolled  in  4'layers 


ra.u$' 

E1.1191J 


SECTION  O.F  SPILLWAY 

FARNHAM 

Mill  Brook,  Massachusetts,  U.  S.  A. 
Completed  1912 


ASHOKAN  DIVIDING  WEIR 
New  York,  U.  S.  A. 
Constr.  1912-1914 

(See  page  103) 


INDEX 


PAGE 

Amount  and  distribution  of  stress  at  any  point  in  a  masonry  dam .  .  197 

Application  of  arch  equation 161 

Application  of  cantilever  equations 1 54 

Arch  and  cantilever  actions,   solution  for  distribution  of  loading 

between 152 

Arch  and  cantilever  analysis,  method  of 143 

Arch  dam 134 

Arch  dam,  limitations  of  the  invesigation  of 142 

Arch  dam  of  rectangular  cross-section 167 

Arch  deflection  equations 152 

Arch  equation,  application  of 161 

Arch  equations,  table  for 163 

Arch,  nomenclature  for  the 148 

Arch  section 136 

Arch  stresses 166 

Arch  subjected  to  uniform,  radial  load,  movements  and  stresses  in. .  221 
Arched  and  straight  gravity  dams,  which  may  be  used  to  best  advan- 
tage    134 

Arched  dam  investigation  outlined 142 

Ashokan  reservoir,  description  of 75 

Ashokan  reservoir,  storage  capacity  of 74 

Atcherley's  analysis  on  stresses  in  masonry  dams 169 

Austin,  Pa.,  dam,  cause  of  failure  of 15 

Austin,  Texas,  dam,  cause  of  failure  of 15 

Baker,  Sir  Benjamin,  experiments  by,  on  dam  stability 180 

Bazin's  experiments  on  sharp-crested  weirs,  vertical  and  inclined 

up-  and  down-stream 126 

Bouzey  dam,  France,  cause  of  failure  of 15 

Buttress  arch  dam 135 

Cain's  consideration  of  stresses  in  masonry  dams 197 

Calculations  for  determining  the  theoretical  cross-section  of  Olive 

Bridge  dam 77 


270  INDEX 


Cantilever  deflection  equations 150,  151 

Cantilever  equations,  application  of 1 54 

Cantilever  equations,  derivation  of 207 

Cantilever,  nomenclature  for  the 145 

Cantilever  stresses 165 

Capacity  of  Catskill  aqueduct 74 

Cataract  dam 10 

Catskill  aqueduct,  capacity  of 74 

Causes  of  failure  of  masonry  dams 35 

Center  of  pressure  on  submerged  surface 22 

Conclusions  from  experiments  upon  two  model  dams 173 

Concrete  and  mortar,  tensile  strength  of 27 

Condition  of  stress  in  masonry  dams,  recent  considerations  of 169 

Conditions  for  stability  of  masonry  dams 35 

Conditions  for  stability  of  vertical  sections  more  critical  than  for 

horizontal 177 

Conditions  under  which  it  is  not  necessary  to  provide  for  ice  pres- 
sure       17 

Constant  angle  dam 138 

Cracks  in  masonry  mass  due  to  temperature  changes 4,  97 

Cross-section,  equations  for  determining 54 

Cross-sections,  analytic  work  in  designing  and  investigating  should 

be  checked  by  the  graphic  method 73,  93 

Cross-sections  of  dams,  dimensions  and  conditions  for  five 12 

Cross-sections  of  existing  masonry  dams 231 

Crushing  strength  of  masonry,  limit  of 2,  13 

Dam  foundations,  effect  of  upward  pressure  in 5 

Dam,  pressure  limits  reached  at  toe  and  heel  of 48,  49 

Dam  stability,  experiments  by  Sir  Benjamin  Baker  on 180 

Dams  of  various  sections,  Unwin's  conclusions  on  investigations  on.  187 

Dams,  width  of  top  of 43 

Danger  of  neglect  of  large  tension  across  vertical  sections 173 

Dangerous  tensions  existing  across  vertical  planes 182 

Deflection  in  the  cantilever,  development  of  equations  for 211 

Derivation  of  the  expression  for  horizontal  (cantilever)  moment 

of  inertia 207 

Description  of  Ashokan  reservoir 75 

Design  of  dams,  formulae  for 31 

Determination  of  angle  <t>n 161 

Determination  of  maximum  and  minimum  pressure  in  a  masonry 

joint ,.  f .....,.,.,.. 25 


»    INDEX  271 

PAGE 

Determination  of  shear  on  horizontal  planes , 183 

Determination  of  stresses  when  down-stream  face  ceases  to  be  linear, 

graphical  solution  for 1 76 

Development  of  equations  for  deflection  in  the  cantilever 211 

Differential  expressions  for  flexure  of  a  cantilever 1 50 

Dimensions  and  conditions  for  five  cross-sections  of  dams 12 

Distribution  of  loading  between  arch  and  cantilever  actions,  solutions 

for 152 

Distribution  of  pressure  in  masonry  joint  subjected  to  external 

forces 22 

Distribution  of  shearing  stress  on  horizontal  joints 171,  183 

Diversity  of  opinions  regarding  uplift  and  ice  thrust 3 

Doubtful  value  of  factor  of  safety 41 

Drainage  channels  to  eliminate  upward  pressure 10 

Drainage  wells  and  galleries 9 

Eccentrically  loaded  joints,  variation  of  pressure  in 24 

Eccentricity  in  a  dam,  manner  in  which  it  is  produced 24 

Effect  of  upward  pressure  in  dam  foundation 5 

Equation  for  deriving  value  of  the  safety  factor,  resultant  at  middle 

third 36 

Equation  for  determining  tension  in  a  joint 27 

Equation  for  finding  the  length  of  any  joint 45 

Equation  for  ice  conditions 59,  61,  63,  70,  115 

Equation  to  determine  depth  of  rectangular  portion  of  dam 45,  46 

Equations  for  determining  a  cross-section 54 

Equations  for  determining  the  length  of  joints  from  the  top  down. .  .  50 

Equations  for  flood  conditions 59,  60,  65,  108 

Equations  for  spillway  dams,  illustration  of  use  of 112 

Equations  for  various  conditions  of  overturning  moment,  spillway 

dam  110-115 

Excluding  water  from  masonry  dams 14 

Existing  masonry  dams,  series  of.  (Indexed.) 231 

Experimental  investigations  of  stresses  in  masonry  dams. .  173,  180,  187 
Experiments  by  Bazin  on  sharp-crested  weirs,  vertical  and  inclined 

up-  and  down-stream 126 

Experiments  by  Bazin  on  weirs  of  irregular  shape 127 

"Experiments  upon  two  model  dams,  conclusions  from 173 

Extent  and  distribution  of  upward  pressure 12 

Factor  of  safety,  doubtful  value  of 41 

Factor  of  safety,  modification  due  to  uplift 37 


272  INDEX 

INDEX 

Failure  of  Austin,  Pa.,  dam,  cause  of 15 

Failure  of  Austin,  Texas,  dam,  cause  of 15 

Failure  of  Bouzey  dam,  France,  cause  of 15 

Falling  sheet  of  water  over  spillway  dam,  shape  of 120 

Falling  sheet  of  water,  velocity  and  pressure  heads  in 123 

Features  involved  in  a  consideration  of  stresses  in  masonry  dams, 

by  Prof.  Cain 198 

JL'  irst  adoption  of  uplift  and  ice  thrust  in  high  masonry  dam  design . .  2 

Flexure  of  a  cantilever,  differential  expressions  for 150 

Flood  and  ice  conditions,  Olive  Bridge  dam 78 

Flood  conditions  combined  with  ice  pressure  in  design 91,  92,  93 

Flood  level  calculations  combined  with  ice  pressure,  method  for ....  58 

Formula  for  a  condition  of  uniform  intensity  of  stress  over  a  joint .  .  23 

Formulae  for  design 53,  59 

Formulae  for  design  of  dams,  development  of 31 

Formulae  for  design  of  spillway  dam 1 10-115 

Formulae  for  investigation 69 

Formulae  for  the  determination  of  water  pressure 21 

Friction  factor,  Levy  and  Rankine's  angles  for 171 

Friction  of  masonry  surface  modifies  thickness  of  sheet  of  falling 

water 132 

Frictional  and  shearing  resistance  of  a  joint 41 

General  conditions  of  uplift,  three 7 

General  formulae  for  imposed  conditions  of  loading,  Series  F 65 

Graphical  solution  for  determination  of  stresses  when  down-stream 

face  ceases  to  be  linear 176 

Gravity  section,  as  a  choice  of  type 136 

Gravity  section,  tabulation  of  results " 68 

Guarding  against  ice  thrust ' 16 

Harrison's  conclusions  as  to  necessity  for  ice  thrust  provision 17 

Hill's  analysis  of  stresses  in  a  masonry  dam 195 

Horizontal  planes,  determination  of  shear  on 183 

Horizontal  planes    passing  through  inner  toe,  tensile  stresses  on 

other  than 193 

Ice  pressure,  conditions  under  which  it  is  not  necessary  to  provide 

for 17 

Ice  pressure  design,  spillway  dam 115 

Ice  thrust  and  upward  pressure •: l 

Ice  thrust,  guarding  against 16 


INDEX  273 


PAGE 

Impossibility  of  tension  on  vertical  planes  near  the  outer  toe 193 

Index  of  existing  masonry  dams 233 

Influence  of  uplift  and  ice  thrust  on  stability  of  masonry  dams 3 

Intercepting  drainage  system,  theory  of 9 

Investigation,  formulae  for 69 

Lateral  strains,  Poisson's  ratio  for 139 

Levy  and  Rankine's  angles  for  friction  factor 171 

Limit  of  crushing  strength  of  masonry 2 

Limitations  of  investigation  of  the  arch  dam 142 

Loading,  general  formulae  for  imposed  conditions  of 65 

"Loading,"  various  conditions  of,  introduced  into  formulae 53 

Manner  in  which  water  pressure  is  exerted  against  submerged  sur- 
face   20 

Marklissa  dam 8 

Masonry  dam,  amount  and  distribution  of  stress  at   any  point 

1         in i95>  197 

Masonry  dam  failures,  precautions  in  design  for  preventing 35 

Masonry  dams,  causes  of  failure 35 

Masonry  dams,  excluding  water  from 14 

Masonry  dams,  recent  considerations  of  condition  of  stress  in 169 

Masonry  dams,  series  of  existing  cross-sections  of 231 

Masonry  dams,  superelevation  of  top  of 43 

Masonry  dams,  water  leakage  through  concrete  of 4 

Masonry  dams,  width  of  top  of 43 

Masonry  joint  subjected  to  external  forces,  distribution  of  pressure 

in 22 

Maximum    and    minimum   pressure  in  a  masonry  joint,  determi- 
nation of 25 

Method  of  arch  and  cantilever  analysis 143 

Method  of  determining  pressure  at  joint  by  resultant  couple 28 

Methods  for  preventing  masonry  dam  failures 35 

Moment   of   inertia,   derivation   of   the   expression   for  horizontal 

(cantilever) 207 

Moments  of  pressure,  resisting  and  overturning 39 

Movements  and  stresses  in  arch  subjected  to  uniform  radial  load ...  221 

Nomenclature  for  masonry  dam  formulae 31 

Nomenclature  for  the  arch 148 

Nomenclature  for  the  cantilever 145 


274  INDEX 

PAGE 

"  On  some  disregarded  points  in  the  stability  of  masonry  dams," 

Mr.  Atcherley's  paper 169 

Olive  Bridge  Dam 74,  96 

Olive  Bridge  dam,  calculations  for  determining  the  theoretical  cross- 
section  of 77 

Olive  Bridge  dam,  flood  and  ice  conditions . 78 

Ottley   and   Brightmore's   investigations   of   stresses   in   masonry 

dams ; 187,  195 

Overfall  or  flood  level  design,  formulae  for  spillway  dam 108 

Overturning  moment,  equations  for  various  conditions  of 108 

Poisson's  ratio  for  lateral  strains 139 

Precautions  in  design  for  preventing  masonry  dam  failures 35 

Pressures  at  toe  and  heel  of  dam  attained 35,  48,  49 

Rectangular  cross-section,  arch  dam  of 167 

Rectangular  portion  of  dam,  equation  to  determine  depth  of 45,  46 

Resisting  and  overturning  moments  of  pressure 39 

Safety  factor,  equation  for  deriving  value  of  the 36 

Safety  factor,  modification  due  to  uplift 37 

Shearing  stress,  on  horizontal  joints,  distribution  of 171,  183 

Shearing  stresses  on  vertical  planes 189 

Simultaneous  equations,  table  for  final 164 

Spillway  dam,  design  of 105 

Spillway  dam,  ice  pressure  design  formulas 115 

Spillway  dam,  illustration  of  use  of  equation  for 112 

Spillway  dam,  overfall  or  flood  level  design  of 108 

Spillway  dam,  shape  of  falling  sheet  of  water  over 120 

Stability  of  dam  from  standpoint  of  vertical  sections  must  be  first 

consideration 174 

Stability  of  masonry  dams,  influence  of  uplift  and  ice  thrust  on 3 

Stability  satisfactory  for  horizontal  sections  may  be  unsatisfactory 

for  vertical  sections,  conditions  for 177 

Storage  capacity  of  Ashokan  reservoir 74 

Stress  variation  in  masonry  joints,  no  exact  law  for 23 

Stresses  in  masonry  dams,  experimental  investigations  of. .  173,  180,  187 
Stresses  in  masonry  dams,  features  involved  in  a  consideration  of, 

by  Prof.  Cain 198 

Stresses  on  vertical  planes,  shearing 189 

Stresses  on  vertical  sections , , 175,  204 


INDEX  275 

> 

PAGE 

Submerged  surface,  manner  in  which  water  pressure  is  exerted 

against 20 

Superelevation  of  top  of  masonry  dams 43 

Table  for  arch  equations 163 

Table  for  computation  of  "  EZ>'s," 162 

Table  for  computation  of  S'2  and  S2 160 

Table  for  final  simultaneous  equations 164 

Table  of  coefficients  for  6,  in  equations  for  D 156 

Table  of  distances  of  origins  above  joints 156 

Table  of  values  for  shapes  of  nappes 122 

Tabulation  of  results  of  arch  dam  analysis , 165 

Tabulation  of  results  of  gravity  design 68,  94 

Temperature  changes,  cracks  in  masonry  mass  due  to 4,  97 

Tensile  strength  of  concrete  and  mortar 27 

Tensile  stresses  on  other  than  horizontal  planes  passing  through  inner 

toe 193 

Tension  across  vertical  sections,  danger  of  neglect  of  large 173 

Tension  existing  across  vertical  planes,  dangerous 182 

Tension  in  a  joint,  equation  for  determining 27 

Tension  on  vertical  planes  near  the  outer  toe,  impossibility  of 193 

Theory  of   dam  stability,   Atcherley's  conclusions  from  investiga- 
tions    179 

Theory  of  intercepting  drainage  system 9 

Thickness  of  sheet  of  falling  water,  friction  of  masonry  surface 

modifies 132 

Treatment  of  uplift 6 

Trigonometric  function  of  <£n 149 

Uniform  intensity  of  stress  over  a  joint,  formula  for  a  condition  of . .  23 

Unwin's  conclusions  on  investigations  on  dams  of  various  sections. .  187 

Unwin's  "  Theory  of  Unsymmetrical  Masonry  Dams  '"' 182 

Uplift  and  ice  thrust,  diversity  of  opinions  regarding 3 

Uplift  and  ice  thrust  in  high  masonry  dam  design,  first  adop- 
tion of 2 

Uplift,  meaning  of 4 

Uplift,  three  general  conditions  of 7 

Upward  pressure  and  ice  thrust i 

Upward  pressure,  drainage  channels  to  eliminate 10 

Upward  pressure,  extent  and  distribution  of 12 

Upward  pressure,  variation  of  c  to  meet  different  conditions  of.  . .   72,  73 

Upward  pressure,  various  ways  of  caring  for 8 


276  INDEX 

PAGE 

Value  of  the  coefficient  C  in  formula  for  discharge 130 

Variation  of  c  to  meet  different  conditions  of  uplift 72,  73 

Variation  of  pressure  in  eccentrically  loaded  joints 24 

Various  conditions  of  "loading"  introduced  into  formulae 53 

Velocity  and  pressure  heads  in  falling  sheet  of  water 123 

Vertical  planes,  dangerous  tensions  existing  across 182 

Vertical  planes,  shearing  stresses  on 189 

Vertical  sections,  stresses  on 175,  204 

Water  leakage  through  concrete 4 

Water  pressure,  formulas  for  the  determination  of 21 

Weir  or  overfall  dams 99 

Weirs  of  irregular  shape,  experiments  by  Bazin  on 127 

Width  of  base  of  triangular  dam  for  varying  heights  and  uplift n 

Width  of  top  of  dams 43 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 
BERKELEY 


Return  to  desk  from  which  borrowed. 
This  book  is  DUE  on  the  last  date  stamped  below. 


MAR  J 1  1950, 
APR  22V 
FEB  17 

JA 


LD  21-100m-9/48(B399sl6)476 


YC  33131' 


UNIVERSITY  OF  CALIFORNIA 

DEPARTMENT   OF   CIVIL    ENGINEERING 

BERKELEY.  CALIFORNIA 


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Engineering- 
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UNIVERSITY  OF  CALIFORNIA  LIBRARY 


